The crack is more obvious in the top picture. In the lower pic there is a crack in the grout near the door frame.
Here are the supplies that I needed. At Lowes, I bought sanded caulk (AKA grout caulk) and a little tool to smooth the caulk out. A got a piece of chipped grout from the bathroom and put it in a ziplock bag and took that to Lowe's with me. The sanded caulk comes in different colors, and I wanted to get the closest color match that I could. The sanded caulk feels a lot like grout when it dries, but it is kida rubbery like regular caulk. We already had the caulk gun and screwdriver pictured.
I used the screwdriver to chip the damaged grout away, then I got a hand vac to clean up the mess. Then I applied the sanded grout. It was a messy job. I kept two old washcloths and and old towel within reach to clean up mistakes as I went. I used nail polish remover to clean up any smudged caulk, and it worked nicely. Here's the finished product:
I had to remove the air vent while I worked, but it just slips in and out.