There's a lot of overlap between 'peep' and 'peek.' If native speakers approve, I would support adding a couple of alternative German translations with 'gucken' for both.
In general, AE seems to prefer 'peek,' especially in phrases: 'No peeking!, 'steal a peek,' 'take a peek,' etc. 'Peep' in the sense of 'spähen/gucken' is still understandable in AE, but my sense is that it survives in our active vocabulary primarily in the phrase 'peeping Tom.'
LEO currently seems to have only
peep - piepen, piepsen; spähen (durch ein Loch, eine Ritze usw.)
peek - nachsehen, spähen
sneak a peek - einen Blick erhaschen
@Muttersprachler: Oxford-Duden has 'peep² - (look through narrow aperture) gucken (ugs.); spähen (geh., veralt., noch landsch.).' Are those caveats about 'spähen' true? And what about 'nachsehen' vs. '(nach etw.) gucken,' the latter of which Pons-Collins gives as the only basic translation for both 'peek' and 'peep' in this sense?
@BE speakers: Oxford-Duden still lists 'peep-bo' in the sense of 'peek-a-boo.' I only know it as a G&S character (and maybe a nursery rhyme -- is that where 'Little Bo Peep' comes from?). Is it really still used? If so, maybe it should be added too, marked [Brit.].