In a nutshell... no.
Get the 100-400... you will be miles ahead of trying to use an extender on the 75-300.
The 100-400 isn't really all that big and heavy. It's very handholdable, until it's extended it's only a little bigger than a 70-200/2.8.
The Canon extenders won't even physically fit onto the 75-300. They have a protruding front element that fits into the rear of the lens and the 75-300 isn't large enough diameter. So if you wanted to try that, you'd have to go to a third party teleconverter.
A 1.4X "costs" you one stop of light and a 2X costs two stops. So even with the weaker teleconverter your 75-300, at best becomes an effective 420mm f8 lens, which will no longer autofocus on your camera (anything less than f5.6) There actually is a work around to this, you fool the camera into trying to focus... but in anything less than ideal light it might fail, and is likely to struggle and be slow to focus all the time.
If you did manage to find a third party teleconverter that fit and either focused manually or managed to get it to autofocus to a limited extent, you will most likely find image quality is... well, to be frank.... crap. For one, in general extenders work better on prime lenses than they do on zooms. And the 75-300 is a less than stellar zoom to begin with. Adding a teleconverter to it, it's likely that image quality will be really poor.
Personally, I wouldn't think it worth the effort... I'd just get a 100-400 or one of the other longer zooms and be done with it.
"Is there an extender that is compatible with this lens ..."
As a 'general' rule, extenders are a poor solution. They are a bandaid approch because nothing gets better when you use one except focal magnification. If that is your sole goal, I suspose you could do it. When you try to use one on a marginal lens, the result will not be spectacular.
Whether one will actually fit, is up to others as I have not tried to do so.
Two problems arise when you use an extender.
First... in addition the extender changing the focal length range based on the extender factor (e.g. a 2x extender means you double the focal length -- which you want), you also have to remember that the same mulitplication has to be performed on the focal RATIO range as well (which you do NOT want but that's a rule of physics so you can't skirt around it). That means an EF 75-300mm f/4-5.6 lens becomes a 150-600mm f/8-11 lens (notice how I multiplied both the focal length range by 2x and the focal ratio range by 2x.)
This creates a big problem because it's generally not possible to get auto-focus to work correctly at f/8... there's simply not enough light (only the 5D III and 1D X can auto-focus at f/8... no other camera body in the lineup that I'm aware of can do that.)
The next problem is optical quality. The extender always compromises the optical quality -- at least a little. This is especially a problem for the 75-300mm because it's already a lens which is a bit mediocer on optical quality. The benefit of the 75-300mm zoom is it's affordability. Every other telephoto zoom in the Canon product line outperforms this lens.
This means that even if it were possible to use an extender with this lens, you'd take a noticeable hit on optical quality and would probably not like the results.
If what you want is a lens that can zoom to 400mm then you're MUCH better off investing in the lens that can natively do that without an extender.
12/05/2023: New firmware updates are available.
09/26/2023: New firmware updates are available.
08/18/2023: Canon EOS R5 C training series is released.
07/31/2023: New firmware updates are available.
05/18/2023: New firmware updates are available.
03/30/2023: New firmware updates are available.