"A lot of people are traveling on the long trains."
Translation:A hosszú vonatokon sok ember utazik.
It sounds a bit unnatural, but it's not wrong. It should probably be accepted.
The reason it sounds a bit strange is that the position in the sentence just before the verb is where the emphasis usually lies, so your variant emphasizes that a lot of people travel on the long trains even though some other mode of transportation may have been expected. This may make sense in the context of a conversation, but out of context it's unnatural.
[Native speaker] Although "sok ember" means "many people", it is considered singular in Hungarian: "sok ember utazik". If you think of it as a group of people, you will recognize that the same thing happens in English, because the group itself is considered singular: The group travels/reads/writes etc.
If you want to use "utaznak", then the subject also must be plural, like: "Az emberek utaznak." = "People are travelling." If you want to express that MANY people are travelling, you can do it like this: "Sokan utaznak." = "Many [people] are travelling."
Hope that I could help you clarify the difference between the two. :)