Yes, the law requires that public schools allow students in private school to participate in extracurricular activities, including sports.
The law regarding participation in extracurricular activities at a public school is the same for homeschoolers and private school students. In either case, there is no specific restriction on which school where they participate, but it does state that it must be in “the student’s school district of attendance or in the student’s school district of residence.” It also allows for participation at a school in a contiguous district if the activity is not offered within their district.
In general, you must choose a single school for all sports that your child participates in, although there may be exceptions. For example, if your child wants to join a swim team, and there is only one school in the district with a pool, they may be able to swim for that school even if they are playing football for their neighborhood school. Be sure to ask your local district about how they handle this kind of situation.
You can read the whole section of the law about this under “Extracurricular Activities” on the Colorado Law page.