"Headache" and "vomiting" are grammatically dissimilar: the first is a noun and the second is a verb form. They need to be similar in order for the sentence to be well constructed. One option is to make both be nouns: "nausea and a headache". Or, you can make them both verbs: "vomiting and an aching head".

You can also say “I vomited and had a headache yesterday.”

There is a few sentences you can make such as... Yesterday I was vomiting and with a headache. Yesterday I was vomiting and I also had a headache.