Actually neither of them is a sentence, there is no verb.

The first is correct. The apostrophe is needed because the grammar and vocabulary skills belong to the students.

First one :) shows possession that it’s their vocabulary and it’s more than one student. I know it’s not a complete sentence but this is correct. This could be a response to someone’s question. “What are you doing?” You respond with the first if you’re writing

Best in most cases is probably just using the singular: ‘Assessing student vocabulary and grammar skills’ (That’s a perfectly concise and clear phrase) It could be followed by, for example: … is done by computer. … was my role at the school. … proved to be useless. … will be time-consuming but necessary. In most cases, it’s understood that there can be more than one student and that each student receives an assessment of his or her capabilities. You don’t need plurals or possessives to convey that information.

The correct sentence is: Assessing students’ vocabulary and grammar skills. In this sentence, the apostrophe and "s" (students') indicate possession. The phrase "students’ vocabulary and grammar skills" means that you are assessing the vocabulary and grammar skills that belong to the students. The correct use of the possessive apostrophe helps clarify the relationship between the students and the skills being assessed.