Hi coldfingersandtoes
My DDs (now Y4 & Y6) have had problems at various points in YR - Y2 finding someone to play with and have felt very downcast about it but we have stumbled onto a reasonable solution by my pointing out what they can be doing on their own to have fun.
Look around the playground - is there a hopscotch grid or similar? Teach your DD how to play and encourage her to play on her own if she can't find something else to do. Encourage her to always include anyone who asks to play - the easiest way to make friends is to be open to play with them.
Is there a climbing frame? - now sometimes there is a rota for playground equipment but remember you can remind them that on Tuesdays they can play on the climbing frame or the bars, etc... - so they know what they can do that day.
Talk to the teachers about what toys/ resources are available in the playground - there may be bicycles, scooters, skipping ropes, balls, etc... which she can play with on her own.
See if she can have permission to draw on the playground with chalk.
See if she can have permission to take out paper and draw during recess or read a book.
see if you can send along a colouring book or similar for her to play with during recess.
My DD2 is in a very strange class with some very pushy & argumentative girls (not awful but very determined to have it their own way) that she finds very hard going. In Year 1 she just decided it was easier not to play with them than to be yelled at or teased. So she started doing a lot of solitary activities - drawing, skipping rope, playing hopscotch, reading, rescuing insects and putting them into flower beds, making designs out of leaves/ flower petals etc... At first she was all by herself but gradually people would ask her what she was doing and even join in. She's now got a nice group of kids who play with her when they want to be a bit more quiet or their usual friend is away for whatever reason - and several kids just play with her regularly anyway now, because there aren't any 'issues'.
At age 4 your DD is still very young and in effect is more likely to be playing alongside other children than with them. I'd talk to a teacher and let them know that you have seen your child wandering about distressed (let them know biting her sleeve is that signal) and ask that if they see her in that state could they assist. Most schools also have playground buddies or areas where you can stand that indicate you're looking to play with someone - so again ask the teachers if something like that exists.
My advice going forward is to accept that making friends takes time & shared experience. Consider having your DD join an after school club next year. Also encourage your DD to have activities she can fall back on if her friend(s) aren't there to play with.
Finally be aware that this may be harder on you than it actually is on her. I know it's hard but try not to show her you're concerned about this. Keep positive about school & recess. See opportunities for fun/ play which can be done with or without other children. And finally encourage your DD to be kind to others and always include them - long term that will win her many friends.
HTH