Canon F1 film advance lever stuck and shutter won't release

This camera is 42 years old. Sounds like a mechanical problem, but I doubt you are going to find any company or individual who knows how to fix it, or can find parts for it, if needed.
Suggest you keep this museum piece as a conversation starter, and invest in a newer model.
If you want to stay with Canon, the Elan 7N and Rebel T2 film cameras are 'only' about 13 years old. I see Elans on eBay starting at $70 (body only).

A Canon AE-1 is not an EOS camera the Canon "A" series is manual focus while EOS is auto focus. First thought In answer to the advance lever stuck or jammed I'm going to assume the camera is in serviceable condition and has not been subjected to water damage or harsh use. If we are dealing with just the advance lever then I'd say that you have over advanced the film at the last frame. I've done this myself after loading a 24 exposure and thinking I had a 36 exposure film. At the bottom of the camera there is a little pin which releases the advance lever and allows the user to rewind the film. Under normal conditions when it's time to rewind the film into the canister this pin offers little resistance to pushing in to release, however depending on how much force the advance lever has been subjected to increases the tension on this pin. Push the pin in and rewind the film.
Second thought is if you have film in the camera and it has not been used fully then it's possible that the film has already been advanced to a new frame and the shutter needs to be activated before the lever will become free and advance to the next frame.
Other problems with the Canon "A" series cameras is they have a tendency to dry out the lubrication and begin to squeak when the shutter is fired. This condition can eventually result is a shutter seizure and give the same advance lever condition.
Those are the three things that come to mind over this lever problem lets hope it's just run out of film.

After taking the first shot, hold down the Multi-Exposure button while cocking the film-advance lever. This cocks the shutter without winding the film. You can continue to do this for as many exposures as you want. Just don't hold down the M-E button while cocking after the last exposure.

The Multi-Exposure button is the button on the front of the camera just below the shutter release button.

The film is either at the end or stuck. The shutter will not release unless the film advance lever has gone all the way and back.
Make a note of the frame counter. Rewind the film and remove it from the camera (When you feel the film to come loose from the receiving spool, stop rewinding to prevent the end of the film to disappear inside the can. This way you can reload it if it was stuck in the middle.) If the film was completely exposed, take it to be developed.
With the empty camera, try to advance the lever and release the shutter. If this works, the problem was the film and not the camera. Everything is fine. If the problem persists, take it to be repaired.
If the film was stuck in the middle, load it again into the camera. With the lense cap on (preferably in a dark room or similar) 'shoot' as many 'empty' frames as you had on the counter plus two more. Finish the film and develop the pictures.
If the film stucks again at the same place, rewind it and develop normally. It was a bad film.
Hope this helped you to solve the problem.

Camera has a bad magnet, the A series were plagued by this.
Repairs will cost more than probably what it's worth.
The AE-1's used an electromagnet system to activate the shutter for a smoother press and less shake from the camera system.

I got lucky and recently purchased an AE-1 program that doesn't have this problem, it seems to be a hit and miss with these.

needs repairing at a qualified shop
when was it last cleaned or serviced ?
if you think the camera is worthy, since its old how much is it worth to you vs a new one

This is likely because the shutter has not been cocked all the way. Return the lens to the fully locked position and advance the film with the lever all the way down to reset the shutter. If it still doesn't work, move the lever to M for multiple exposure, make sure the back is aligned and trip the shutter release. Then fully **** the shutter again by pressing the advance lever down fully and again until it is reset. Then try to remove the lens again.

In most film cameras, a battery is not necessary to manual advance film using the lever. Your Canon AV-1 is no exception. I have owned two cameras from the same group, the AE-1 and the Canon A-1. Neither required a battery for manual film advance.

Make sure the ring around the shutter release button is set to "A", otherwise the shutter won't release and the film lever won't work correctly. If the ring shows an "L", the shutter release is locked.

Here's a graphic of the batteries that can be used in your AV-1:

Do you press the shutter button to release the lever for the next photo? That's how you advance the film from the exposure to light, when first insterting the film.