Are 1D Textures Supported in WebGL yet?

Why do you need 1D textures? Just make a 2D texture N pixels wide and 1 pixel tall.

var tex = gl.createTexture();
gl.bindTexture(gl.TEXTURE_2D, tex);

// 3x1 pixel 1d texture
var oneDTextureTexels = new Uint8Array([
    255,0,0,255, 
    0,255,0,255,
    0,0,255,255,
]);

var width = 3;
var height = 1;
gl.texImage2D(gl.TEXTURE_2D, 0, gl.RGBA, width, height, 0, gl.RGBA, gl.UNSIGNED_BYTE,
              oneDTextureTexels);

Either generatemips or set filtering so no mips are needed

gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_MIN_FILTER, gl.LINEAR);
gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_WRAP_S, gl.CLAMP_TO_EDGE);
gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_WRAP_W, gl.CLAMP_TO_EDGE);

Sample it with with 0.5 for y

uniform sampler2D u_texture;
varying float v_texcoord;

void main() {
  vec4 color = texture2D(u_texture, vec2(v_texcoord, 0.5));
  ...

Here's a sample using 1D textures. It uses the dot product of a typical lighting calculation to look up a value from a 1d ramp texture to shade the objects.

In direct answer to your question. There will be no 1D textures in WebGL because WebGL is based on OpenGL ES 2.0 and OpenGL ES 2.0 does not support 1D textures. Neither does OpenGL ES 3.0 nor 3.1. I'd be surprised if they didn't remove 1D textures completely when they merge OpenGL and OpenGL ES

WebGL 1.0 is based on OpenGL ES 2.0 which does not support 1D textures. The Texture Objects section in the WebGL specification reflects this by only having texImage2D and compressedTexImage2D methods.

You can use a texture with a height of one instead.