Learn Dutch: Ik lach, want ik ben blij. - I am laughing, because I am happy.

Dutch does not have a separate '-ing' form (continuous tense) like English does. Ik lach translates to both 'I laugh' and 'I am laughing' depending on what is happening in the context.

How does this sentence relate to the question word waarom?

This sentence is a natural answer to a waarom (why) question. If someone asks 'Waarom lach je?' (Why are you laughing?), you use want (because) to introduce your reason.

Does want change the word order of the second part of the sentence?

No, want is very friendly for beginners! It simply links two normal sentences together. Ik lach and ik ben blij both keep their standard subject-verb word order.

I learned another word for happy, gelukkig. Can I use that here instead of blij?

You could, but blij fits better here. Blij means you are cheerful or glad in this specific moment, which matches the quick act of laughing. Gelukkig is usually used for a deeper, long-term happiness or feeling lucky in life.