Can you experience vaginal itching and irritation after you've finished taking a round of clindamycin for 10 days, as perscribed by my doctor?

Yes indeed. Probably a yeast infection, which often is triggered by antibiotics that suppress normal vaginal bacteria, allowing yeasts (normally present in the vagina, rectum, etc) to overgrow and cause inflammation. Unless it clears up on its own in a few days, it probably will need treatment with a vaginal or oral anti-yeast medication. Discuss with the doctor who prescribed the clindamycin.