I guess it depends on the online casino. Most will shuffle the deck after each hand played. If they from a single deck, and it's casino-style with multiple players or you're playing multiple hands, and you're sitting at third-base (the last seat to be dealt), I *guess* you might be able to sometimes have an edge by couting the cards on the table and, if the count went high enough, playing index plays as appropriate. But unless they're the more exotic plays it won't be too beneficial as it won't let you adjust your bet higher with a higher count of course.
If the casino doesn't shuffle after each hand and instead shuffles once and does it casino real world style, then you could count. You'd have to shop around. Shuffling after each hand in real casinos cuts down on casino profits because it takes so much time to do and thus they play less hands per hour and thus less profit, so they of course don't do it; in an online computer setting, simply calling your shuffle algorithm before each round is dealt takes no time at all. So I'd imagine most if not all online casinos do this.
BUT, if you play basic-strategy to a T, and the house only has a 0.40% edge over you, you're likely to have some good 'winning streaks' as opposed to someone that just plays by the seat of their pants. Check out the Basic Strategy Engine on this site and generate a rule table for the particular rules of the game you are playing.