How would you find out if a Will has been submitted to probate court without asking the family member in charge?

If a will that requires probate is not entered for probate, the will's validity has not been established. Any distribution pursuant to that will may be challenged.

Probate is a matter of public record. Most courts now have online case search. For example if you google "tarrant county probate court records search" one of the top results will take you to a page where you can search by party name.