Both are correct, but they mean different things. "I saw her this year" is the simple past tense. It refers to a completed action. Examples: "Have you ever seen Katy Perry in concert?" "Yes, I saw her this year." This is a specific, completed event - seeing a Katy Perry concert. "When did you last see your mother?" "I saw her this year, in February." This is a specific, completed event - the last time you saw your mother. "I have seen her this year" is the present perfect. It means that from the beginning of the year, until now, you saw her at least once, maybe more than once, and the general action of "seeing her" may not be completed. Examples: "Have you ever seen Katy Perry in concert?" "Yes, I have seen her twice this year." This speaker has been to two Katy Perry concerts this year, and he may go to another one. "Have you seen your mother lately?" "Yes, I have seen her this year." This speaker has seen his mother at least once this year, and he will probably see her again. Here is a page I found using Google that talks about the difference: http://www.perfect-english-grammar.com/present-perfect-or-past-simple.html

Both are correct, "saw" sounds more natural.