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.
The distinction is not important in this case. Everyone means every person, without exception and anyone means any person, without discrimination.
Usually a tournament would be "open to everyone" or "open to anyone".
To me there is a marginal difference that "open to everyone" sounds slightly more friendly or more welcoming than "open to anyone".
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.
"every" as in "each and every" is a reference to elements in a set and can be taken as the intention to include all elements in the set; "any" is more a universal concept and implies no distinction at all.