Can I purchase storage on my iPhone with an iTunes gift card?

Yes, sure.

Check here -> What you can buy with your Apple Gift Card or Apple ID balance

What you can buy with your Apple Gift Card or Apple Account balance – Apple Support (UK)

See also: How to redeem your Apple Gift Card or App Store & iTunes gift card

How to redeem your Apple Gift Card or App Store & iTunes gift card – Apple Support (UK)

When redeeming a gift card, the amount will be credited to your (1) Apple ID balance and (2) you need to set your Apple ID balance as your default payment.

(1)

See -> Check your Apple ID balance

Check your Apple ID balance – Apple Support (UK)

(2)

Change, add or remove Apple ID payment methods

Change, add or remove Apple ID payment methods – Apple Support (UK)

If you update your payment method or add funds to your Apple ID balance, It would take about 1 to 3 business days in order to process.

If you're still unable to update your payment method, Contact Apple Support.