I was this child, it was crippling.
I used to go to things like playcentres with my cousins, all fine, but if mum dropped me and my sister somewhere I didn't know anyone I'd sooner sit at the side counting down the minutes until pickup time.
I was the late august baby who started school at 4 and a week, having had a SAHM so no daycare or preschool.
I think it was so overwhelming I withdrew more and never really learned to be different, I was also ashamed of how difficult I found it to make friends so would hide in the bathrooms at lunchtimes during high school than be seen to have no friends.
I didn't really learn how to make friends until my 30s, it's a skill to be learnt, and is also something you can talk about, give ideas and explain how to go about it.
Being shy does hold you back.
I would see if the kids have a more extroverted friend maybe from school that you could nurture a friendship with, yours will learn social skills from them as well as confidence.