They do.it's a shame that IPSC and IDPA and other shooting organizations don't have rimfire divisions. They would bring in a lot more members, and in times like these, numbers are everything.
IPSC has rimfire rifle competitions. It's very big in UK, since it fits within their firearm laws.
The Smith and Wesson M&P-15-22 is by far the most popular rifle they use for it.
Ironically; That rifle is banned in Canada.