Any of the Canon lenses that will handle fast action in low light will have no issue handling the same action in bright sunlight so you have to decide/buy based upon performance in low light and this means a fairly wide aperture lens that will be heavier and more expensive.
My most used sports lens is a Canon EF 70-200 f2.8 and it will be with me at every sports event. I use 2 or 3 camera bodies depending upon the sport BUT a 1DX III with the EF 70-200 f2.8 will always be the primary piece of gear.
Lacrosse and hockey are both fairly difficult sports to shoot because of the speed at which the action moves across the playing area. It makes it more difficult for the photographer to anticipate where the action will shift and because of this I have to rely more heavily on the camera to "save me" in some situations which means I depend upon the camera to focus quickly when I misjudge what is going to happen next. Regardless of camera body, a wide aperture lens will provide for faster focus performance because of the additional illumination it provides to the sensor.
If the price of a new 70-200 f2.8 is beyond the budget you set, consider used AND for sports I don't consider IS absolutely necessary. I used the 70-200 f2.8 non-IS I bought back in 2005 for well over a decade and didn't miss IS when shooting sports because the required shutter speed is high enough for sports action that camera shake isn't a concern. IS is definitely nice for other uses and if it fits your budget, then a 70-200 f2.8 with IS is desirable but it is certainly possible to get a lot of good sports photos without an image stabilized lens.
A longer focal length is definitely nice for sports but if you need a fast lens (and you will except for the best illuminated professional and college D1 facilities), then the price gets very high even used. My second most used lens for sports after the 70-200 f2.8 is the 400 f2.8 but it is definitely not a budget lens. The "super zoom lenses" from Sigma and others that cover 150-600 provide good optical performance but they aren't suitable for low light and the focus acquisition speed is marginal. Lately I have been making more use of my EF 200-400 f4 with built in switchable 1.4X extender (instantly switches back and forth between native or 280-560 f5.6) and I love it for daytime sports but even with a 1DX III I still prefer my 400 f2.8 at night because its performance seems magically good at times.
I would start with the 70-200 f2.8 which should net you some good shots if you can get reasonably close to the action. If you can't get close to the action, none of the "super zoom" lenses with that body will provide good results in reduced light. I use a Sigma 150-600 as an all purpose hiking lens but even for daytime sports, I don't find its focus performance to be fast enough. I did use a Canon EF 400 f5.6 prime for many years for daytime soccer and it was an excellent lens but not suitable for low light/fast action use.
But you can count on the 70-200 f2.8 to get the job done whether the sun is shining brightly or it is the nastiest cold pouring rain at night imaginable 🙂
Rodger
