Volksturm tended to be issued captured or obcelete rifles and damn all ammunition, and would be highly unlikely to have a PPSH.
Eastern front vet possibly, but funnily enough I've found a couple of references which suggest that captured smg's were issued as standard to 'trench raiding parties' for individual actions where the additional firepower would be useful. Seems, from admitedly limited evidence, to have been common practice as it is not treated as out of the norm when I've seen it mentioned. As airsoft tends to be short range skirmishes I should just go for it if I were you, whether you want to be a 'vet' or a bog standard Heer chappie. Perhaps even a nice bog standard Gebirgsjager 
... not that I'm biased you understand. 
IIRC captured PPsH were the second most common issued German SMG after the MP40, both the unmodified and the modified ones. Obviously much more common on the Eastern Front but the odd one made its way West as units were transferred, so fine for regular army types.
Volkssturm are more likely to have long Mausers and be supported by MG08/15s and rechambered Lewis guns! I don't see why they couldn't have a PPsH as a battlefield pickup though, Volkssturm in the east general put up more of a fight than their mates in the west.
Cheers
Martin
"Mistakes in the initial deployment cannot be rectified" Helmuth von Moltke
Toys: AGM MP40, Cyma M1A1, TM M14/G43/SVT40, TM VSR/K98, SnS No. 4, ASG Sten, Ppsh.
Arnhem3,Gumrak,Campoleone
What size are you. I have a set of m40's going up for sale. Size 40 jacket and 34/36 trousers. Never been used. Litzen and brest eagle included.65 quid posted.
Go for it
no worries
In which case, militaria-net would be a good option.
You've got nothing to ein, zwei, drei, vier
