Is 'with' needed/

No. it's not needed. It actually doesn't make sense.
She fed her pet carrots
Carrots were fed to her pet
fed her pet carrots ,she did

"With" is unnecessary.

"With" would imply the method of feeding.

She fed her pet with a spoon.

Or an additional ingredient:

She fed her pet carrots with medicine.

Or could be used to describe her state of mind:

She fed her pet carrots with great joy.

Can I say "I fed my baby with medicine" when I'm giving her the medicine through her mouth?

I don't think 'fed' is the correct verb. Then what verb should I use? Or should the sentence be rephrased?

Many thanks.

Rephrase.

Formally: The child's medicine was administered orally.

Informal: I gave the baby her liquid antibiotic mixed with applesauce.

I dosed the baby with the antibiotic the doctor prescribed.