Another word for "apply"?

Probably not the best word choices.

For one thing, the verb apply has several meanings, beyond the one you intend. A naive user might well believe that your service would be "applied to them" (somehow) if they press the "apply" button. If you haven't been astonished yet at what naive users can believe, get ready.

For another thing, the noun application also has a number of meanings, of which one (the one that gets shortened to app and shows up as a file type .app in some systems) is central to any computing usage. The fact that that isn't what you mean probably won't help; I suggest you don't give this confusion any room to grow.

You need something that does more than merely imply a communication request instead of a piece of software. I'd go for Submit or Post for the button label, and Status of your Submission instead of Application Status.

"Apply" and "Application status" are correct in this context.

I would stick with them.