I’m looking for a word that comes from the old Germanic language. Not just the word Penis since that is derived from Latin.
Looking at the trusty Indo-European etymology dictionary, there used to be a German term, but its use has died out.
The root for penis is
Old Indian: pásas- n. `penis'
Old Greek: péos n. `das männliche Glied'
Slavic: *opesъkā (Czech opeska `praeputium')
Germanic: *fás-Vl=, *fís-Vl=
Latin: pēnis, -is m. `Schwanz; männliches Glied'
Russ. meaning: penis
Drilling down into the Germanic roots leads to
Proto-Germanic: *fásVl=, *físVl=
Old High German: faselt 'penis'
Middle High German: visel, vësel st. m. 'penis'; vasel st. m. 'der fortpflanzung dienendes männliches vieh, zuchtstier, -eber u. dgl.', st. n. 'das junge, die nachkommenschaft; gezücht, gesinde'
Low German: fisel 'penis'
Translating the German explanation:
Male animal used for breeding, bull, hog, etc; also young, litter, progeny
I can only speculate whether the colloquial terms Piesel and pieseln, the latter meaning urinating, are related as the link doesn't say. Seeing as the shifts in Germanic languages generally went from *p to *f (one of Grimm's Laws IIRC), I'd rather suspect that these are borrowed from the Latin/Romance.
There isn't one.
That, of course is an exaggeration, but there isn't a candidate for your requirements among the most commonly used neutral synonyms. That isn't uncommon, and it's not necessarily to do with any taboos either. German doesn't have a native word for 'leg' either, so 'Bein' is used, which properly means 'bone'.
To be sure, there are many, many synonyms, in fact perhaps more than for most nouns. But there isn't one single-word synonym that is clearly germanic and clearly the standard usage. Perhaps the closest thing is the expression "das männliche Glied" (the male member).
"Schniedelwutz" is a german word meaning "penis".