Can the cost of additional storage be taken out of my iTunes balance? I do not wish to enter a credit card?

Yes.

You don't need to do anything. iCloud Storage, like other iTunes and App Store purchases, is automatically charged to your iTunes credit balance before it requires a credit card.

As long as there is enough to cover the cost of the subscription plus any applicable taxes it will be charged to your balance.

See here: iCloud storage plans and pricing - Apple Support

Accepted payment methods for iCloud storage upgrades include credit cards, debit cards, and your Apple ID balance.

and here: How App Store and iTunes Store purchases are billed - Apple Support

You don't need to do anything as I said.

As long as you have enough credit balance in the Apple ID you use for iCloud, it will use that automatically.

See here for how to redeem an iTunes Gift Card onto your account: Redeem App Store & iTunes Gift Cards and content codes - Apple Support

or add funds to your Apple ID from Apple Pay / Wallet: Add funds to your Apple ID - Apple Support

You select the new storage size, and then tap on Buy on the top right corner.

It may take a moment for it to register.

Details here: Buy more iCloud storage - Apple Support

Restarting your iPhone should not necessary, since the storage is not directly related to your iPhone.