Learn Italian: Cercavo un balsamo per i capelli biondi. - I was looking for a conditioner for blonde hair.

This is the polite imperfect (imperfetto attenuativo). Using the past tense softens your request, making it sound less direct and demanding to the shop assistant. English does the exact same thing: you walk up to a clerk and say 'I was looking for...' even though you are still looking right now.

In English we use the word 'balsam' differently. Is balsamo a false friend here?

Yes, in the context of washing your hair, balsamo always means 'conditioner'. If you are looking for lip balm, you would ask for burrocacao or balsamo per le labbra.

Why is capelli plural when English just uses the singular 'hair'?

Unlike English, which treats hair as an uncountable singular mass, Italian treats the hair on your head as countable and plural (i capelli). Because it is a masculine plural noun, the adjective modifying it must also be masculine plural, which is why we use biondi.

Could I drop the article and just say per capelli biondi?

You will actually see per capelli biondi (without the i) written on product labels, as advertising language often drops articles. However, in spoken Italian, it sounds much more natural to include the definite article just like in this sentence.