Learn French: La plupart des clients demandent une amélioration du service. - Most of the customers ask for an improvement of the service.

With the phrase la plupart de (most of), the verb agrees with the noun that comes after it, not with the word plupart itself. Since clients is plural, the verb takes the plural form demandent.

I thought phrases of quantity like beaucoup just took de. Why is it des clients here?

You are right that many quantity words use a bare de (like beaucoup de clients). However, la plupart usually includes the definite article because it means "most of the". Here, de + les (the) contracts into des.

Why isn't there a word for "for" after demandent?

In French, the verb demander translates directly to "to ask for" or "to request". It takes a direct object, meaning you don't need to add a preposition like pour. You just say they request an improvement: demandent une amélioration.

Why is it du service at the end instead of just de service?