Definitely check the bed even though the timing seems off since you've had it two months -- still, I would inspect it carefully. If you got it from a store that sells a lot of imported products, there is chance that the wood is infested. I know people who have had to throw out inexpensive wooden or wicker products that were imported because the items had bugs in them. (Not "bedbugs" but other insects that were shipped here along with the furniture.) You could try removing the bed for a few weeks and seeing if the bites then stop; if they .do stop, the problem was something in the bed that you couldn't see but that was there, nevertheless. .
The other possibility is, as others said, spider bites, and in spring spiders often move indoors if it's been wet where you are. Put out spider traps (you can get them at any hardware store or Home Depot type of store) as well as investigating the bed.Spider bites can be harmless but certain types can be serious, though not for your son so far. I might check with an exterminator and ask if there are issues in your area with biting spiders in homes.