This would require them to be in the front line early. I think it's more likely that somebody was randomly there and saw the opportunity. If you look at the videos most seem to be surprised to be in some of low manner. If you have some operative in there you increase chaos by creating some small fire or something ...
Would they have had to be there early? Russian embassy isn't that far away and I think rioters were in there plenty long (wasn't it a couple hours?) for someone to walk over, pay 20 dollars for a trump flag, and walk in unnoticed. I wouldn't be surprised if our embassies and CIA outposts have people ready to take advantage of such situations in other countries.
You have to get through the crowd, and you have to be there before somebody else stole it and in time before police manages to lock down critical offices. Quite some effort.
Compare it to a random person suddenly standing in front of an unprotected laptop from those "bad politicians" ...
