Can I "daisy-chain" LED strip lights to my bathroom exhaust fan and have them work on the same light switch?

Since your LED strips are powered by a 5V DC the correct answer is:

No, you can't directly add DC equipment to an AC circuit.

However, you can add a 110 V AC outlet to the switchable part of the circuit that the fan is on and plug in what ever adapter you were planning on running your LEDs off of from that.

Assuming you want these LEDs on the ceiling near the center of the room, you probably want to use a ceiling box (for lights) instead of a standard switch box so that any future owner, or change in taste will allow you easily to convert to another fixture. along with a cover like this: They also sell these in decora or single outlet style:

Alternately, (and what I'd probably do) in place of a ceiling box, you could install a recessed light fixture, then use a socket to outlet adapter and plug your 5V DC transformer into that:

I see that in your comments you are worried about flickering. This shouldn't be a problem, unless the circuit is overloaded (if so you need to run a new line from your breaker box), or had a ton of compact florescent bulbs on it (they dirty the power by only pulling during the maximum voltage differential).