You're running into a camera "buffer" limit. You may find that you can solve your problem just by shooting JPEG instead of RAW and you don't need to upgrade anything.
The camera has some amount of internal memory. When you take a photo, the photo does not actually go straight to the card... it goes to that temporary memory. The camera will then start writing the image to the card BUT... in the meantime the camera is available to take another shot (even though it's not done writing the last shot.)
This memory buffer (temporary memory) can hold several shots... but it's not unlimited. The actual number of shots it can hold depends on the size of the images that you are taking (not very many shots if you shoot RAW, lots and lots of shots if you shoot JPEG.)
Here's the test results for your camera:
Canon EOS 550D (Rebel T2i / Kiss X4 Digital) In-depth review: Digital Photography Review
Notice that in RAW mode you can shooting 5-6 frames before the buffer is full. If you switch to JPEG mode you get "100+" (which means they gave up once the camera got to 100 frames before it's not realistic that anybody would need more.) While "100+" frames sounds really great... keep in mind the continuous shooting framerate on this camera is 3.7 frames per second.
Here's what you'd get if you "upgraded" to a few different newer bodies then the one you have.
T5i only gets 12 frames of JPEG... but it's shooting them at 5 frames per sec.
60D can shooting about 58 frames of JPEG and it's shooting them at 5.3 frames per second.
7D is basically built FOR these sports/action shooting situations, can shoot JPEG frames at 8 frames per second and keep this up for over 320 frames (about 40 seconds).
70D can get a good 40+ shots and it's shooting at 7 fps.
You can see that the T5i will seem like a "downgrade" but that's really because it's frame rate is faster. The 60D is even faster still but it's internal performance and memory buffer are sized that it really outperforms the Rebel class bodies (and it should... that's part of why it's a mid-range body and not an entry-range body). The 70D is the latest in the mid-range and it shoots noticeably faster than the 60D (not to mention having a better focusing system.) But the 7D... well it's basically purpose built for the job. Almost nothing beats it... almost.
My 5D III (6 frames per second) has "unlimited" in JPEG mode. The camera can process and write the images to the CF card faster than 1/6th of a second which means the buffer will never fill up.... but that's an expensive camera. And likewise the 1D X blows the doors off the 5D III. But these are $3500 and $6000 bodies. (the 1D X is blazingly fast).
If you _really_ need a long continuous frame rate, the 7D or 70D would be good options (and frankly I'd probably lean toward the 70D because it's only 1 fps slower and it's doubtful you'd need more than 40 continuous frames).
Three pieces of advice:
1) Shoot in JPEG mode if what you want is the ability to shoot a LOT of continuous frames without the buffer filling too quickly.
2) Buy the FASTEST class card you can.
3) Anticipate! Knowing you're playing a game of getting the shots without overflowing the memory buffer... don't start shooting TOO soon. If you know the sport or type of action you are shooting, you get a sense for when something photo-worthy is about to happen and you want to capture that "decisive" moment. If you start shooting too soon then you'll start filling that buffer.