You shouldn't need an ADC, although many small RF modules generally output 'analogue', since the transmission is digital, you only really need to look for 'high/low'.
Best bet is to find a ground reference, most likely connected to the 0v of the battery terminal of the receiver. Then you can put your analyzer probe on each pin of the RF module to see what happens. Perhaps doing this first with a multimeter, so you can see which pins are power. A photo of its guts would be nice
I guess the problem here is that the transmitter only puts out a transmission every so often, so once you've found your 'data out' (which might be labeled on the PCB!) you would have to wait until the transmitter spits out a packet. Thankfully, if you have a logic analyzer (or even a cheap software one for a PC) they generally have a 'trigger', which means you can just arm it, and leave it - once theres a change on the probes input (be it low -> high, or high -> low) it'll start recording. a 10khz sample rate should be more than fast enough.
Now, so far I've been talking about reverse engineering by probing the circuit, which works for ANY system, but its a lot of work and a hassle. Its always preferable to first check out what IC's are on the board. Although many scrub off the part numbers, or use programmable microcontrollers (ie: all firmware) occasionally you'll see a chip with a part number, which may be of some use. Googling the datasheet of this tells you everything you need to know about it. This method saved me hours of work cloning a broken DVD remote control - since they used a well known IC', and its datasheet described the protocol perfectly. Rarely happens, but you never know.
Also, this is jsut how I would do it, if anyone else has better idea's, please post. You'll not only help the OP, but also those in a similar situation!