The Doctor's companions (whoever you choose to regard as companions) cannot regenerate merely by virtue of being his companions. A few, though, could regenerate because they had that ability anyway:
- Melody Pond (a.k.a. River Song & generally regarded as a companion) was of human ancestry but could regenerate because of the circumstances of her conception. She regenerated twice (once in "Day of the Moon" & once in "Let's Kill Hitler"). In "Let's Kill Hitler", however, she used up all her remaining regenerations saving the Doctor from the poison with which she had tried to kill him. She had, therefore, lost the ability to regenerate before (in her timeline) she became his companion.
- Romana (also generally regarded as a companion) was a Time Lord. She could & did regenerate (in "Destiny of the Daleks").
Any other Time Lord whom you count as one of the Doctor's companions would likewise have the ability to regenerate, provided they hadn't previously used up their regenerations.