Do I need to capitalize "modernism" or "modernist"?
Difficult to be sure without context, but you probably do, by analogy with Cubism, Surrealism, Romanticism, Classicism and so on.
If you are writing about the arts (the term is also current in biblical studies and theology), I would not capitalize modernism.
The names (and derivative adjectives) of some artistic movements are, as Barrie England tells you, properly capitalized: Futurism, Epic Theatre, Socialist Realism, Art Deco, Neue Sachlichkeit. These are all narrowly defined in critical history; each identifies a specific group of artists, body of works, artistic posture or objective; in many cases the artists themselves explicitly adopted the name.
Other such names (and derivatives) are, as Zairja tells you, capitalized to distinguish a narrow use from a more general one: Renaissance, Baroque, Archaic. Note, however, that such capitalization may not extend from a name to a derivative, or vice versa, if the distinction is not necessary: impressionist painting is ambiguous, but impressionism is not.
Modernism, like romanticism and classicism, falls into neither of these categories: modernism will not be confused with the modern, and it is not (except perhaps in architecture) the name of any generally recognized and narrowly defined school or group or movement or body of works, or of specific artistic principles or objectives. (On the contrary, the term has been so widely applied, to so many artists, that I gravely doubt whether it means anything useful!)
You might, by way of analogy, consider political terms. Communism is capitalized when it refers to Marxist-Leninist-Maoist ideologies and regimes, but not otherwise; liberalism and conservatism and socialism are capitalized only when they refer to the platforms and principles of parties which call themselves Liberal or Conservative or Socialist.
So unless you can define a specific group of artists, with specific principles or techniques or objectives, who call themselves or were called in their heyday Modernists, I wouldn’t capitalize the word. I wouldn't even use the word without first carefully defining exactly what I meant by it; and even then, I would expect three quarters of my reviewers to argue with my definition.
It doesn't appear that you need to. Several online sources, including Wikipedia, are inconsistent with their capitalization (this includes impressionism, post-modernism, surrealism, etc.).
In published sources, including Herschel Chipp's Theories of Modern Art and a couple of Norton anthologies I have, modernist remains lowercase. You can capitalize for the sake of clarity when the lowercase spelling could infer an ambiguous or general definition (e.g. romantic poetry vs. Romantic poetry, modern art vs. Modern art).
In practice there doesn't seem to be a consistent answer. Modernism and Postmodernism are capitalized by many writers, and by many they are not. Modernism, unlike modernity, represents a specific viewpoint regarding art even though it includes a variety of art movements such as Cubism, Futurism, and so on.