What is the purpose of 系's constant use in ギャル語? Is it just filler or supposed to give a certain feeling or meaning?
系(kei) essentially means type or style. For example, there are many fashion styles which use it such as Mote-kei — translated as pretty-style but it literally means “attractive-style”.
系 (kei) is often used in ギャル語 (gyaru-go) as a way to add emphasis or express a sense of belonging to a certain group or trend.
In this sentence
系(kei) is used more loosely to mean something related to or of the same type as the preceding noun "買い物" (shopping).
The phrase "モールとか買い物系と？" can be translated to "like shopping in the mall or something?"
You mentioned in your comment that she might be thinking about some guy, so there could be additional context which shifts the meaning, but from this single frame I'd take it to mean she's trying to decide how to spend her day: "(I could do) something like shopping at the mall. I wonder if there are any events..."