。
I'll contact the pharmacy from our end.
Literal
Pharmacy [to-に][topic-は] this-side [from-から] contact-[ておく].
Stacked particles には (に + は) mark 薬局 both as the destination of 連絡 and as the topic — 'as for contacting the pharmacy...' こちらから ('from our side') is the polite way of saying 'we'll handle it, not you.' The ~ておく auxiliary signals the action is being done in advance or as preparation, a nuance English typically leaves unspoken.