I am on state felony probation with a suspended sentence.
One of the conditions of my probation is that I am to have no internet access.
What you’re describing is not uncommon, though not all grants of felony probation come with a ban on using the internet. You can calendar the matter for a probation modification form the judge, or hire an attorney to do so. The likelihood of that petition being granted would depend primarily on the underlying conviction, in my opinion. But to answer your question directly, yes you can be deprived of certain First Amendment rights as a term of probation.