There's a phrase that has been around photography a long time:
Gear doesn't matter.
But that's only half the truth. The fuller truth is:
Gear doesn't matter... until it does.
What it means is that when the technical capabilities of your gear are not up to the task for the shots you want to capture, then and only then will the gear matter.
Applying the above to your case: both the camera body and the lens you are using are fairly limiting for the type of photography you want to do.
- No matter what lens you hang on it, the AF system of the Rebel T3/1100D is going to be severely limiting for wildlife photography. Frame rate is also quite limiting. Since taking plenty of frames in burst mode is often one way to deal with poor AF system performance, the two combined are kind of a double whammy.
- No matter what camera body you hang it on, the Sigma 50-200mm f/4-5.6 is fairly weak optically, particularly at the long end. It's also slow to focus. Even if it were sharper at 200mm, it would still be fairly focal length limited for most kinds of wildlife photography.
You can replace the lens first. You can replace the body first. But you're probably not going to get where you want to go until you improve both.
If you're on an extremely limited budget, you'll need to consider used gear that is a generation or two (or three) old.
The original 7D, for all of its faults, is capable of getting pretty good wildlife/action photos if you're willing to deal with its shot-to-shot AF inconsistencies. You'll have a lower "keeper" rate than with more consistent AF systems on later cameras, but you will be able to get some "keepers" with a 7D. The 70D has the same AF system but is not quite as built to stand up to punishment as the 7D. But a lightly used 70D may be a better deal than a worn out 7D. Either can be had for a lot less than a brand new 7D Mark II or 80D.
In terms of lenses, there's not as much price difference between new and used copies of the same models. I'd probably go ahead and buy new unless I stumbled across a really good deal from someone I trusted.
Sometimes the Canon USA refurb store has good promotional prices on refurbished lenses. I've bought an EF 17-40mm f/4L, EF 100mm f/2, and EF 135mm f/2L from the Canon refurb store. They all looked and performed like new when I got them. But some of the most desirable lenses are rarely in stock at the refurb store. I'd been waiting forever for a good deal on the 135/2 when I finally gave up and bought a 100/2 (which is also seems to be out of stock more often than not). A few months later a 135/2 showed up in stock and I snapped it up before it was gone.
In the U.S. there are a handful of used gear sellers with very good reputations and acceptable return policies. KEH, based out of Atlanta, is one. The "used" department at B & H is another. Lensrentals.com sells used gear via their in-house lensauthority.com.
In terms of specific lenses, the EF-S 55-250mm f/4-5.6 IS STM is about as good optically as the EF 70-300mm f/4-5.6 IS when used on a crop body. Either are significantly better than the EF 75-300mm f/4-5.6 III which you should avoid.