I sympathize with what Megan is trying to do, because we have been that family that moves a lot. And our kids are always desperate to have friends to invite to their birthday parties. But when you just moved into a new state at the end of September and your kid's birthday is a month later, it's really hard to know people to invite!
If she'd accept advice from somebody who has been there, done that, what I'd do is to put off the birthday party as much as possible. Maybe have a small family recognition on the day of (a small cake or treat, family presents). Give your child a few more weeks to attend church, daycare, preschool, AWANA, whatever program/class/group your child is in, and let them get to know people.
Then invite just a *few* kids, those your child is closest friend with. I've usually heard that a reasonable number of kids to invite is one for each year of age your child is turning. So a 3-year-old would invite 3 friends, etc.
She might also consider, when she talks to the people and hands them an invitation, that she's just moved to the area, her child is really excited about having playmates come to a birthday party, etc. That might help them understand why she's inviting them when they don't know her well.
Of course, if the party is really soon or already happened (since you mentioned your child is sick and wouldn't be able to come), it might be too late. But if your child was the *only* child coming and nobody else shows up, she might try a "do-over." Another option could be to see if it could be a birthday playdate, with just your child and hers, if she just hasn't had enough time to get her child involved in groups so there would be kids to invite.
My heart hurts for that little one, if the toddler *is* old enough to wish for friends to come to her party, as I know my daughters really struggled with this. They've each developed differently because of all the moving. One is more of an introvert and entertains herself really well. Another is really, really top-notch at remembering people's names and making friends. We were out of town this weekend and went to a new church, and she made a friend, visited with the friend after church, and wants to go to that church every time we're in that area now, so she can play with her friend again.