Try Squatting and pushing down. Make a hook with your finger after you insert it into your vagina. OMG i cringed when I read that. That happened to me before. I had to lay on my bed and reach around until i felt the tip of the gauze. You did use gauze, correct? You are supposed to take the garlic wrap it in gauze or cheesecloth and then tie the end with unflavored dental floss.
If you didnt do that then I can recommend clamping down really hard with your vaginal walls and hope that it comes out. There is a fool proof way to get it out if it was just the clove and I feel a bit embarressed saying it : spank the monkey. It will force your vagina to tighten and release around the garlic and the fluid should help move the garlic out. If you can not get in to see your GYN then you should go to the Emergency room if you dont want to take it out. Also do you have tweezers or in some cases tongs. If you place KY jelly on then you can reach in and get them out. If you have a significant other or someone you trust you can ask them to help you get it out.
I googled garlic stuck in vagina and its actually a pretty common situation. Here are the results, Hope it helps, at least on the bright side youll be fighting off some more yeast.