Most Soto Zen sources that I've seen advise daily Zazen, preferably when first rising and before going to bed. However, few make reference to a length of time. The only one I remember says half an hour. At this point I feel like 15 minutes would be a serious challenge (I am a beginner). Is this a problem on my end, or could I perhaps start with 15 and work up? If so, at what rate should I increase?
If you have a Sensei or Roshi, ask them. If not, I would recommend you to start with a length which is challenging but not impossible to manage. From time to time you can do a longer meditation. In example, on a weekend sit 4 times 15 minutes with 5 minutes kinhin between. You'll find this a valuable experience. Don't just stay with a specific time without changing: no challenge means no progress. Zen meditation is about interrupting, its hard work. If you feel relaxed, work harder.
There is no official recommendation because every teacher recommends it different. In example Kodo Sawaki emphasizes Zazen and he would most likely recommend stricter practice than modern teachers.
I would definitely recommend you to find a meditation group (sangha) as the sangha is also part of the way. Questions like you asked can't be answered easily, especially by people like me who never met you. Additionally you might get theoretical answers by people like me who are not competent enough to guide you (I am not a sensei, just a student). Remember, zen is a way, not just a quantified meditation block.
However, if you don't have a sangha close (or just crazy people in it) it's better to start somehow instead of doing nothing. So I would suggest the 15 minutes and work up and keep reading old masters. I recommend sawaki specifically to you because of the question you asked.
I'd like to recommend Christian's answer: assuming that "15 minutes is a serious challenge", it might be a good experience to find a teacher and/or other practitioners to practice with. I find that many activities are much easier in a group: I find it easier to do what other people do, when they do it, and to continue (not stop) because they are continuing (not stopping), and to stop when they stop.
There are two types of practices: the formal SITTING and WALKING, and the informal the activities in daily life. Sit as long you feel comfortable than stand up walk up and down until you feel you can sit again. And so on. During the day find some moments where you do nothing just sitting still, slightly concentrating on the breathing so that the mind has an anchor. The most important meditation is during all your activities. If you cannot be meditative while you are driving a car or answering a phone call your meditation is not worth much. And how to do it? Whenever you can slow down. Walk slowly, eat slowly, prepare your coffee slowly, even speak much slower than you are accustomed.
You are free in this choice. I recommend periods from 5 minutes to 40 minutes. It is better to practice 5 minutes daily than 15 minutes only once a week.
If you practice more and the sitting doesn't hurt to much you stretch your sitting time naturally.