I'm not going to add terribly much to what other people have said, but here's my 10pence worth:
DEF not enough notice, esp at this time of year. My dd's birthday is in grim autumn/winter, and people very rarely refuse her invites (esp when the kids were younger - they're all teens now) and that was cos the parents were so grateful to get rid for their kids for a couple of hours when the weather's grim. At this time of year, people tend to do more stuff because the weather's generally better (not today where I live, btw!). We give invites out generally with 3 - 4 weeks notice and/or DD talks at school to her mate and/or I mention it to other mums.
You do, as others have said, need to make more effort. No one else has suggested this, but I'd suggest joining the PTA. That'll help you get to know other mums and give you a 'role' at events (bound to be a summer fete soon, and they'll be grateful for ANY offers of help - you can just litter pick or something if you're feeling shy). I can not emphasize enough how having a 'role' helped me.
If you want to change the date, suggest you speak to the entertainer first and explain and see whether they can be flexible. Then to everyone else, say the entertainer had to cancel (agree reason with entertainer in advance so you don't get caught out).
If you do decide to go ahead, and there's some good suggestions upthread here for if you do, I'd also add "the Chocolate Game" - always a winner!)
And, yes, sorry, but start organizing some play dates. Even informal, like "it's a lovely day today, anyone want to join us at the park?" (and have a few essentials in a big bag - snacks, water, suncream).
Good luck. Understand why you feel rubbish, but as another poster said - you CAN do this 