What is a word that can be used like "socialist," but for capitalism?

Those who support the ideology of socialism are called "socialists".

However, those who support the ideology of capitalism are not called "capitalists". Instead, those who own capital are called capitalists.

Is there a word I can use to describe those who support the ideology of capitalism?

Yes, capitalist. The premise of your question is faulty.

While it's true that the word can mean someone who owns capital, that's only one sense of the definition of the word.

From Merriam-Webster:

noun
2 : a person who favors capitalism

adjective
2 a : practicing or advocating capitalism
// capitalist nations
2 b : marked by capitalism
// capitalist period of history

Note, too, the definition of capitalism:

: an economic system characterized by private or corporate ownership of capital goods, by investments that are determined by private decision, and by prices, production, and the distribution of goods that are determined mainly by competition in a free market

As with any word that has more than one meaning, context determines the meaning in any particular case.