My daughter is in 3rd grade, and has lots of friends at school, but no one ever calls for play dates. She is fine and never asks, but I know that if I wanted her to have any play dates, I would have to initiate it. When I do, it happens. So, try to put your self to the side and set up a play date. My daughter's one friend that I would set play dates with lived in a small apartment. We live in a big house with a yard. However, I too was always self conscious about inviting over because we were never big presents and toy people, so my kids did not have as much as others, so I always thought kids would be bored here). Anyway, my daughter loved going to the aparment. And now, when the kids come here, they find what to do, so I have learned that they will fend for themselves and have a good time, they are kids. Once you get to know the parents, they will likely drop the kid off, so it will just be the kids playing and you will be amazed how they find things to do.
It may be uncomfortable for you at first, but just try it. If after three times you still can't do it, then ok, you can stop, BUT, I think you will see that you were holding yourself back unnecessarily.
Go for it (for her). (you can also (and not as an alternative, but as a second location, set up a play date at a park, or a day at a kids place).