How to deal with a clingy child?

My youngest just turned 3 and in the last month she’s been acting different, I’d say since school started. We have a very hard time taking her to daycare in the mornings, she will throw a fit and beg to stay home with us (while we both have to work out of home). Nothing has changed in her routine except her brother is in kindergarten now and no longer there but they weren’t even in the same group so idk.

It’s really hard because we don’t want to take her to daycare when she’s upset but like… I gotta get to work child lol… A couple weeks ago I left while she was still crying and she ended up vomiting and I had to pick her up.

I think that’s also separation anxiety but even at home when we are not going anywhere she always wants to be held or cuddled; and it makes it hard to function, especially with two other kids that don’t necessarily understand what’s going on. I really don’t know what to do.