About three months ago, an agent that I had queried responded with a request for the full manuscript of the novel that I had written to her about. It's now been nearly three months, and I haven't heard anything back. What's a reasonable amount of time to wait before pinging her and inquiring about the status of the manuscript? What's an appropriate way in which to query?
It's perfectly reasonable to follow up at this point. I've read numerous agents' and editors' opinions on this, and never have I seen anyone claim you should wait longer than three months (one such example).
I'd suggest you follow up with an email, reminding the agent of your previous correspondence and asking politely when you can expect a response. Email nicely rides the line between slow (snail mail) and too immediate (phone).
Generally 90 days is the standard wait time before sending an inquiry. You might want to check the agent's website and see if they have something in their submission guidelines that says how long you should wait before sending an inquiry.
Generally a simple email asking the status of your submission is acceptable. It may be that they just haven't gotten to it yet, or they replied and the reply got lost in cyberspace. Just remember to be polite.