To picnic is a recognized verb.
You can say go picnicking. It has the same meaning as going on a picnic or having a picnic. Note that some collocations may be more or less common in certain dialects - like some people say have a shower, while others say take a shower.
When you're inviting someone to go with you on an activity, you can substitute go with come. So, you could say come picnicking with me, or come on a picnic with me. To me, this sounds friendlier than go. Come with me emphasizes that it is with the speaker. Go with me emphasises the place you are going to.