I recently drove my Toyota Camry through a very large puddle at a very low speed. The car still runs fine with no strange noises. I am worried that it may have acquired some water in the engine. Should I get this checked out to prevent any future problems?

What make you think you got water damaged?
If the car runs fine, then you probably didn't. Very lucky!
Just in case, take your air filter out and check for water in that. Inspect the air cleaner box for water.

And, please, next time: go around. You may not be so lucky next time. If the car was lost, you could be too...

It doesn't sound like any damage was done. One of the main concerns is if sensors, alternator, or the ECM short out. Allow everything to dry out and, if the car runs fine a day or so after driven through the water, it's unlikely you'll have any problems.

Your engine computer would know if something was up. It would count misfires and send a signal to the check engine light. A flashing light while driving means that a single or multiple cylinder misfire is happening.
A steady on light means it saw a problem twice and it's time to take notice of the problem.
Of course when water enters the engine in most unlucky cases you hear about, this all happens so fast, the damage is done. You are dead in the water so to speak!
Cars aren't boats I tell my clients. Turn around and go another way!!

Even if the engine and all it's vitals were spared, this time, you could have still flooded the passenger compartment and cause enough damage to total the car.