Open for "anyone"? Or open for "everyone"?

The distinction is not important in this case. Everyone means every person, without exception and anyone means any person, without discrimination.

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.

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".

"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.