Of course it does, with a frequency of about 1 per million. Getting a full-blown case of COVID, back when it was the original, alpha, beta or delta strains that caused far more serious disease than the currently dominant omicron strain, caused the same clotting problem in about 20 per million. This data has been around for over a year and a half, but the FDA has gone back to its sluggish ways, so they finally noticed.
Back in the depths of the pandemic, even leaving aside preventing serious respiratory disease, even thinking only about blood clots, getting the vaccine was the right call -- it cut your risk by a factor of 20.
Now it's not so clear. I haven't seen any data on clotting problems from omicron.