"Feel" is a linking verb and we often use adjectives after liking verbs.

We normally use adjectives after "feel"

For example, Do you still feel hungry? (source)

But we can use a noun after "feel", too.

For example, Stop exercising if you feel any pain. (source)

Now, "vomit", "throw-up" and "puke" are nouns

Can we say "I feel vomit/puke/throw-up when I smell pork fat"?

But, "vomit", "throw up" and "puke" are also verbs, and we can convert them into adjectives by adding "ing" to become "vomiting", "throwing-up" and "puking"

Can we say "I feel vomiting/puking/throwing-up when I smell pork fat"?

Some suggests to put "like" after "feel", "I feel like vomiting/puking/throwing-up when I smell pork fat", but I am not so sure.

"Feel" is a linking verb and we often use adjectives after liking verbs.

We normally use adjectives after "feel"

For example, Do you still feel hungry? (source)

But we can use a noun after "feel", too.

For example, Stop exercising if you feel any pain. (source)

Now, "vomit", "throw-up" and "puke" are nouns

Can we say "I feel vomit/puke/throw-up when I smell pork fat"?

But, "vomit", "throw up" and "puke" are also verbs, and we can convert them into adjectives by adding "ing" to become "vomiting", "throwing-up" and "puking"

Can we say "I feel vomiting/puking/throwing-up when I smell pork fat"?

Some suggest to put "like" after "feel", "I feel like vomiting/puking/throwing-up when I smell pork fat", but I am not so sure.