Japanese relative clauses never need particles.
First off, as others have stated,
プレゼント買う心配がありません means "(I) don't have to worry about buying presents" or literally "(I) have no worries about buying presents"
If we swap 心配 with 必要, then we get
"I don't need to buy presents." or literally "I don't have the need to buy presents”.
「プレゼント買う」is basically modifying clause simlar to a gerund meaning “buying present(s),”
Usually gerunds are created by adding の at the end of the verb like
プレゼントを買うのが大好き Meaning “(I) love buying presence.”
But の is not needed when making the modifying clause as the first part of the relative clause. Here are some other examples.
プレゼントを包む心配ありません。 (We) don't have to worry about wrapping the presents.
プレゼントを買う夢を見てた。 (I) had a dream about buying presents.
プレゼントを買った弟がいません。 My younger brother who bought the present isn't here.
The last example takes a relative pronoun in English which is the bane of English Learners in Japan. Here's an article with many examples where Japanese try to figure out how to use them.
You also mentioned missing articles, but the を Is really implied in your sentence. However it is omitted verbally very often.