My puppy eats the decrative bark from the planters and then throws it up. Is this bad for her

What kind of bark? Some contain dyes and chemicals.
Is it the big chunk kind and he's just gnawing on it or little and eating the whole piece?

If your pup is throwing up the bark, then obviously her body cannot digest it.

You need to keep her away from the trees. Its not healthy of anything to continue to throw up.

Yes it is bad for her.

Not only are some barks treated with chemicals but they can also splinter and cause perforations to your dogs esophagus or stomach.
They can also cause bowel obstructions.

Best to get some high garden edging and stop her from getting in there.