In this situation, we could use the verb form with an -s, instead of the -ing form, if we add the word who:
Anyone arriving after the start of the performance will not be admitted until the intermission.
Anyone who arrives after the start of the performance will not be admitted until the intermission.
Everyone signing up before the deadline will receive a free t-shirt.
Everyone who signs up before the deadline will receive a free t-shirt.
I can see where this would be confusing, though, because that's not always the case with plural pronouns:
Everyone eats cake for their birthday.
Not everyone sees the forest through the trees.
The key is that the sentence is really structured like this:
Everyone .. does something.
and we are dealing with some qualifying clause between those two sentence parts.
For example:
Everyone .. will dance the chicken dance.
becomes either:
Everyone who attends the wedding reception will do the chicken dance.
or:
Everyone attending the wedding will do the chicken dance.
One more example:
Nobody sitting in the family section is allowed to drink alcohol.
Nobody who sits in the family section is allowed to drink alcohol.
Anyone (arriving) after the start of the play is not allowed in until the interval
. Admittedly, the previous example was made up but thought it could make the point without having to look for the latter one. Native speakers resort to "it sounds right/natural/awkward", but what about foreigners?