This example is running in WebGL2 and should work in most browsers. You can check the WebGPU examples here.

shader_material_2d.rs:
//! A shader and a material that uses it.

use bevy::{
    prelude::*,
    reflect::TypePath,
    render::render_resource::{AsBindGroup, ShaderRef},
    sprite::{Material2d, Material2dPlugin, MaterialMesh2dBundle},
};

fn main() {
    App::new()
        .add_plugins((
            DefaultPlugins,
            Material2dPlugin::<CustomMaterial>::default(),
        ))
        .add_systems(Startup, setup)
        .run();
}

// Setup a simple 2d scene
fn setup(
    mut commands: Commands,
    mut meshes: ResMut<Assets<Mesh>>,
    mut materials: ResMut<Assets<CustomMaterial>>,
    asset_server: Res<AssetServer>,
) {
    // camera
    commands.spawn(Camera2dBundle::default());

    // quad
    commands.spawn(MaterialMesh2dBundle {
        mesh: meshes.add(Rectangle::default()).into(),
        transform: Transform::default().with_scale(Vec3::splat(128.)),
        material: materials.add(CustomMaterial {
            color: LinearRgba::BLUE,
            color_texture: Some(asset_server.load("branding/icon.png")),
        }),
        ..default()
    });
}

// This is the struct that will be passed to your shader
#[derive(Asset, TypePath, AsBindGroup, Debug, Clone)]
struct CustomMaterial {
    #[uniform(0)]
    color: LinearRgba,
    #[texture(1)]
    #[sampler(2)]
    color_texture: Option<Handle<Image>>,
}

/// The Material2d trait is very configurable, but comes with sensible defaults for all methods.
/// You only need to implement functions for features that need non-default behavior. See the Material2d api docs for details!
impl Material2d for CustomMaterial {
    fn fragment_shader() -> ShaderRef {
        "shaders/custom_material_2d.wgsl".into()
    }
}
shaders/custom_material_2d.wgsl:
#import bevy_sprite::mesh2d_vertex_output::VertexOutput
// we can import items from shader modules in the assets folder with a quoted path
#import "shaders/custom_material_import.wgsl"::COLOR_MULTIPLIER

@group(2) @binding(0) var<uniform> material_color: vec4<f32>;
@group(2) @binding(1) var base_color_texture: texture_2d<f32>;
@group(2) @binding(2) var base_color_sampler: sampler;

@fragment
fn fragment(mesh: VertexOutput) -> @location(0) vec4<f32> {
    return material_color * textureSample(base_color_texture, base_color_sampler, mesh.uv) * COLOR_MULTIPLIER;
}