You need proper stroke order to write Kanji with a pen. You want to write Japanese with a pen a lot if you want to learn to read written Japanese because it builds 'muscle memory', i.e. an intimate, up-close-and-personal knowledge of the structure of the components of Kanji. This helps learning to read, a lot. Learning to read Japanese just by staring at printed characters is a lot harder; it's totally doable but it really gets hard beyond a few frequent and graphically salient characters.

If you want your handwriting to look acceptable, then proper stroke order is an absolute must. When you go further and learn cursive and grass script, then you'll get to learn multiple stroke orders per character (and hundreds of conventionalized doodles for components and whole characters).