Is the tournament open for "anyone" or "everyone"?
There's a subtle distinction, as noted that's likely unimportant -- unless you're at all concerned about subconscious reactions from speakers of American English.
"Anyone" does have some negative connotations and usages attached to it that people don't necessarily think about:
- "Anyone can do that."
- "They'll take anyone."
For your purposes either would probably be fine, but "everyone" will be perceived as more inclusive.
If you are suggesting you would be happy with one (any one) response, which could hypothetically be the case if the condition is that 'there are no restrictions: anyone can participate', then you can use anyone. You don't care who comes.
On the other hand, as may be more likely, if you would like as many as possibly can to participate, you'd be saying everyone. You do want everyone to come.
This is based on how the reader would more generally perceive.