As you suggest, the following sentence does sound awkward.
かぜをひかないために、このサプリメントを飲んでいます。
The reason is that かぜをひかない is not something you actively do, and therefore, not suitable as a purpose.
The sentence below with ように, which denotes a desired state rather than a purpose, sounds much more natural.
かぜをひかないように、このサプリメントを飲んでいます。
If you still need to use ため, you should insert する as below so that the adverbial clause describes what you actively do to achieve a desired state, rather than the desired state itself.
かぜをひかないようするために、このサプリメントを飲んでいます。
However, this distinction becomes less strict in a noun-modifying clause making かぜをひかないためのサプリメント much more acceptable, although some might find it a bit informal and still prefer かぜをひかないようにするためのサプリメント.