Nonwithstanding what Merkel actually said, the sentence as you wrote it is is grammatical. Der can be used as a substitute for der, welcher, although that sounds rather old fashioned, even more so with the subordinated clause put at the beginning -- the formulation reminds me of biblical language. And that's what makes it sound wrong in a context like this,...
I think your transcription must be erroneous: I get Wer unnötige Begegnung vermeidet... when I Google. Admittedly, Merkel's pronunciation of the w is somewhat less than clear (video at appropriate time).
Then this is a subjectival clause: the wer... clause is the subject of hilft.
"Whoever avoids unnecessary contact helps all, who..."