Is there a way to print from ios and android phones to CAnon IR-5353?  An app? settings on the network?

If your Canon Printer has Air Print, you should be able to print

https://www.canon.lt/support/consumer_products/apple_airprint_guide/

I print from my iPhone to an HP Printer with Air Print.

Those large printers do not have air print so I do not know any way you can solve this with the given printer.

You can put an HP with air print on your network and print to that

There may be ways to achieve what you want .. with AirPrint anyway

If the Canon IR-5353 is shared on the network you may be able to use a PC on your network as an intermediary (a print server) by installing AirPrint software .. the PC needs to have WiFi and be on the same LAN as the iOS Devices

If you've an Apple PC on your network you can install Printopia which turns the mac into an AirPrint server
https://www.decisivetactics.com/products/printopia/

If you've only Windows then
- You install the BONJOUR Print Services for Windows (You can install iTunes or just the Bonjour services alone)  
- AirPrint Activator IOS5 v2.6 (or some alternative AirPrint Servers for Windows see links below)
- Go into your Devices and Printers in Windows and share the printers you want available
- Enable the Windows Guest account (or you can use an Administrator account)

https://www.tecklyfe.com/tech-tip-how-to-enable-airprint-for-any-printer-in-windows/
https://tweaks.com/windows/46495/enable-ios-airprint-support-on-windows/

Both when the phone is on the company wifi and on the LAN and also when out of the office on another wifi or cellular data?

This can potentially work for 2 of the 3 cases (not the cellular data scenario). It would be easiest if the corporate wireless and wired networks are the same subnet (this assumes the printer is on it also).

I recommend against allowing remote printing to a printer. Devices should be connected to a VPN first. And there are plenty of VPN clients available for mobile devices.