Learn Finnish: Sauma ja vetoketju ovat rikki. - The seam and the zipper are broken.

The word rikki is an indeclinable adverb, not a true adjective, so it never changes form to match the number or case of the subject. It stays exactly the same whether one thing is broken or many things are.

If rikki cannot be inflected, how would I say broken zippers if I needed to use it right before a noun?

You have to switch to its true adjective counterpart, rikkinäinen, which can take all the normal endings. To say broken zippers as a subject, you would use the plural form: rikkinäiset vetoketjut.

What are the component words that make up vetoketju?

It is a compound of veto (a pull or traction, derived from the verb vetää, to pull) and ketju (chain). So a zipper is literally a pull chain.

Besides a physical seam in clothing, does sauma have any other common meanings?

Yes, in spoken Finnish, sauma is often used figuratively to mean a chance or opportunity. For example, you might hear someone say Sillä ei ole mitään saumaa, meaning he doesn't have a chance or he has no shot.