I will show you how you can make the RasPi a router. I will use systemd-networkd for reasons. For routing we need two subnets. For my example I assume your wifi network has the subnet 192.168.10.0/24 and the subnet the printer is using is 192.168.1.0/24. The printer accepts an ip address from a DHCP server.

For reference I use Raspbian Stretch Lite 2018-06-27 updated with sudo apt update && sudo apt full-upgrade && sudo reboot.

For detailed information look at ⁽²⁾. Here only in short. Execute these commands:

pi@raspberrypi: ~$ sudo -Es
root@raspberrypi: ~# mkdir -p /var/log/journal
root@raspberrypi: ~# systemd-tmpfiles --create --prefix /var/log/journal #ignore warnings (*)

root@raspberrypi: ~# systemctl mask networking.service
root@raspberrypi: ~# systemctl mask dhcpcd.service
root@raspberrypi: ~# sudo mv /etc/network/interfaces /etc/network/interfaces~
root@raspberrypi: ~# sed -i '1i resolvconf=NO' /etc/resolvconf.conf

root@raspberrypi: ~# systemctl enable systemd-networkd.service
root@raspberrypi: ~# systemctl enable systemd-resolved.service
root@raspberrypi: ~# ln -sf /run/systemd/resolve/resolv.conf /etc/resolv.conf

^{(*) You will get one or two confusing warnings "...Cannot set file attribute..." This are not errors and doesn't matter in this case.}

Setup wpa_supplicant with this file and your settings and enable it. You can just copy and paste this in one block to your command line beginning with cat and including EOF (delimiter EOF will not get part of the file):

root@raspberrypi:~ # cat >/etc/wpa_supplicant/wpa_supplicant-wlan0.conf <<EOF
country=DE
ctrl_interface=DIR=/var/run/wpa_supplicant GROUP=netdev
update_config=1

network={
    ssid="[email protected]"
    psk="verySecretPwassword"
}
EOF

root@raspberrypi:~ # chmod 600 /etc/wpa_supplicant/wpa_supplicant-wlan0.conf
root@raspberrypi:~ # systemctl disable wpa_supplicant.service
root@raspberrypi:~ # systemctl enable [email protected]

root@raspberrypi:~ # cat > /etc/systemd/network/04-eth0.network <<EOF
[Match]
Name=eth0
[Network]
Address=192.168.1.1/24
DHCPServer=yes
IPForward=yes
[DHCPServer]
PoolSize=2
EOF

root@raspberrypi: ~# cat > /etc/systemd/network/08-wlan0.network <<EOF
[Match]
Name=wlan0
[Network]
Address=192.168.10.60/24
#Gateway = ip from your internet router
Gateway=192.168.10.1
EOF

To get routing complete working you have to set a static route in your internet router so it can find the route to the printer. On most internet router you can set a static route but how to do that varies from model to model. It's up to you to find it out. For example your RasPi wlan0 interface has the static ip address 192.168.10.60. Then on your router the gateway (next hop) is 192.168.10.60, destination network is 192.168.1.0/24 (or 192.168.1.0 netmask 255.255.255.0).

That means for the internet router: "send all packets belonging to subnet 192.168.1.0/24 (destination network from AP) to the next router on my subnet, the RasPi 192.168.10.60 (gateway). It knows where to go on."

socket://192.168.10.60:9100

On MS Windows machines it should go similar but I'm not familiar with Windows. Look where and how to set connections to network printer in the printer driver with jetdirect (I hope) to printer 192.168.10.60 and port 9100.

I will show you how you can make the RasPi a router. I will use systemd-networkd for reasons. For routing we need two subnets. For my example I assume your wifi network has the subnet 192.168.50.0/24 and the subnet the printer is using is 192.168.1.0/24. The printer accepts an ip address from a DHCP server.

Tested with
Raspberry Pi OS (32-bit) Lite 2020-08-20 on a Raspberry Pi 4B updated at 2020-09-18.
Updates done with sudo apt update && sudo apt full-upgrade && sudo reboot.
Here you can find the last tested revision for previous Raspbian versions.

For reference I use Raspbian Stretch Lite 2018-06-27 updated with sudo apt update && sudo apt full-upgrade && sudo reboot.

Just follow to Use systemd-networkd for general networking. You can use section "♦ Quick Step". Then come back here.

Setup wpa_supplicant with this file and your settings for country=, ssid=, psk= and enable it. You can just copy and paste this in one block to your command line beginning with cat and including EOF (delimiter EOF will not get part of the file):

root@raspberrypi:~ $ sudo -Es   # if not already done
root@raspberrypi:~ # cat >/etc/wpa_supplicant/wpa_supplicant-wlan0.conf <<EOF
ctrl_interface=DIR=/var/run/wpa_supplicant GROUP=netdev
update_config=1
country=DE

network={
    ssid="[email protected]"
    psk="verySecretPwassword"
}
EOF

root@raspberrypi:~ # chmod 600 /etc/wpa_supplicant/wpa_supplicant-wlan0.conf
root@raspberrypi:~ # systemctl disable wpa_supplicant.service
root@raspberrypi:~ # systemctl enable [email protected]
root@raspberrypi:~ # rfkill unblock wlan

root@raspberrypi:~ # cat > /etc/systemd/network/04-eth0.network <<EOF
[Match]
Name=eth0
[Network]
Address=192.168.1.1/24
DHCPServer=yes
IPForward=yes
[DHCPServer]
PoolSize=2
EOF

root@raspberrypi: ~# cat > /etc/systemd/network/08-wlan0.network <<EOF
[Match]
Name=wlan0
[Network]
Address=192.168.50.60/24
#Gateway = ip from your internet router
Gateway=192.168.50.1
DNS=84.200.69.80 1.1.1.1
EOF

To get routing complete working you have to set a static route in your internet router so it can find the route to the printer. On most internet router you can set a static route but how to do that varies from model to model. It's up to you to find it out. For example your RasPi wlan0 interface has the static ip address 192.168.50.60. Then on your router the gateway (next hop) is 192.168.50.60, destination network is 192.168.1.0/24 (or 192.168.1.0 netmask 255.255.255.0).

That means for the internet router: "send all packets belonging to subnet 192.168.1.0/24 (destination network from AP) to the next router on my subnet, the RasPi 192.168.50.60 (gateway). It knows where to go on."

socket://192.168.50.60:9100

On MS Windows machines it should go similar but I'm not familiar with Windows. Look where and how to set connections to network printer in the printer driver with jetdirect (I hope) to printer 192.168.50.60 and port 9100.

[1] Raspberry Pi WiFi to Ethernet Bridge for a server?
[2] Howto migrate from networking to systemd-networkd with dynamic failover
[3] man systemd.unit - overriding vendor settings

root@raspberrypi:~ # cat > /etc/systemd/network/04-eth0.network <<EOF
[Match]
Name=eth0
[Network]
Address=192.168.1.1/24
DHCPServer=yes
IPForward=yes
[DHCPServer]
PoolSize=2
EOF

root@raspberrypi: ~# cat > /etc/systemd/network/08-wlan0.network <<EOF
[Match]
Name=wlan0
[Network]
Address=192.168.10.60/24
EOF

(Work in progress. Will try a better solution)

For reference I use Raspbian Stretch Lite 2018-06-27 updated with sudo apt update && sudo apt full-upgrade && sudo reboot.

root@raspberrypi:~ # cat > /etc/systemd/network/04-eth0.network <<EOF
[Match]
Name=eth0
[Network]
Address=192.168.1.1/24
DHCPServer=yes
IPForward=yes
[DHCPServer]
PoolSize=2
EOF

root@raspberrypi: ~# cat > /etc/systemd/network/08-wlan0.network <<EOF
[Match]
Name=wlan0
[Network]
Address=192.168.10.60/24
#Gateway = ip from your internet router
Gateway=192.168.10.1
EOF

This has the advantage that you can use any service the printer provided. You are not restricted to one port. But if you have no access to the internet router or cannot set a static route there then you can use port forwarding (see next section).

Port forwarding

Thanks to @RalfFriedl for the suggestion in his answer. I will show in detail how I would it use in this setup.

To use port forwarding requires that the printer supports the forwared port. We use port 9100 for raw printing. You can check it with nmap 192.168.1.2 on the RasPi. You must get at least the open port:

9100/tcp open  jetdirect

This is almost suported by all printer and mostly named jetdirect. For example you can select this connection on the unix printing system CUPS with AppSocket/HP JetDirect when you install the printer driver on a computer and set it for this example to:

socket://192.168.10.60:9100

On MS Windows machines it should go similar but I'm not familiar with Windows. Look where and how to set connections to network printer in the printer driver with jetdirect (I hope) to printer 192.168.10.60 and port 9100.

There is a unit to start and stop wpa_supplicant for the wifi connection. We simply add to it the iptables rule so it will be set and deleted with the wifi connection. Create a drop in file ⁽³⁾:

rpi ~$ sudo systemctl edit [email protected]

In the empty editor insert these statements, save them and quit the editor:

[Service]
ExecStartPost=/sbin/iptables -t nat -A PREROUTING -i wlan0 -p tcp --dport 9100 -j DNAT --to-destination 192.168.1.2
ExecStopPost=/sbin/iptables -t nat -D PREROUTING -i wlan0 -p tcp --dport 9100 -j DNAT --to-destination 192.168.1.2

Check with

rpi ~$ systemctl cat [email protected]

Restart the service and check if port forwarding is set:

rpi ~$ sudo systemctl daemon-reload
rpi ~$ sudo systemctl restart [email protected]
rpi ~$ sudo iptables -v -t nat --list

That's it.