I have no problem doing this is 8-bit per channel RGB by using an to load a PNG but this doesn't work with 16-bit per channel data since there aren't any "standard" browser supported image formats which are 16-bit.
I don't think the main browsers natively support any 16-bit/channel image format at the moment.
One way to achieve the same effect would be to create two PNG images, one with the top 8 bits of each colour channel in the image and one with the bottom 8 bits. Then bind the images as two textures and combine the values in your shader, e.g.
highp float val = texture2d(samplerTop8bits, tex_coord) * (256.0 / 257.0);
val += texture2d(samplerBottom8bits, tex_coord) * (1.0 / 257.0);
(Note: you need highp precision to represent your data correctly in a 16-bit range)
Another method is only possible if floating point textures are supported in your target browser(s). You would, in the browser, combine the two PNG images into a floating point texture then access that normally. This may not be any faster and will probably use twice the amount of texture memory.