When Jenica Andersen felt the tug for a second child at age 37, the single mom weighed her options: wait until she meets Mr. Right or choose a sperm donor and go it alone.

The first option didn’t look promising. The idea of a sperm donor wasn’t appealing, either, because she wanted her child to have an active father, just like her 4-year-old son has. After doing some research, Ms. Andersen discovered another option: subscription-based websites such as PollenTree.com and Modamily that match would-be parents who want to share...