I'm reading the relevant laws in Virginia and they're very scarce on details as to the limits of the possible deferral conditions. They may be unilaterally imposed by the court, but most commonly they are agreed to by all parties, including the accused.
One interesting detail from the general text about any deferral is that once you agree to it, you waive your right to appeal a conviction if you're deemed guilty for violating the terms. That strikes me as relevant because it seems like a pretty significant right to give up if you "agree" to it. Fascinating legal question, honestly. I want to learn more
Adding more: agreeing to search without probable cause or notice seems to be a very common probation condition, which is fairly blatantly unconstitutional in general, but it flies for probation because you would technically consent to it because it's preferable to the 'punishment'. No way to be certain, but I feel strongly that internet restrictions are at least possible
Editing once more: We all know at this point about Packingham v North Carolina, which establishes that a North Carolina law banning sex offenders from the internet was unconstitutional. But the argument primarily rests on the breadth of the law, saying essentially, in my reading, that the uses of the internet are far too varied for a ban to legitimately serve the state's interests to protect the public or rehabilitate the criminal.
However, Illinois still explicitly lists social media bans as acceptable conditions of parole.
https://codes.findlaw.com/il/chapter-730-corrections/il-st-sect-730-5-5-6-3.html This has been challenged following the lead of Packingham but has failed to go anywhere repeatedly. Virginia law is not Illinois law, but this shows that severely restricting Chris's internet activity is absolutely possible