Proto-Germanic is commonly reconstructed as lacking a short 'o' phoneme. The subsequent development of short 'o' can be looked at by tracing developments in Gothic, German and Yiddish.
Biblical Gothic, attested from the 5th century, is close to the Proto-Germanic state. There is no 'o' phoneme in the native Germanic word stock although 'o' is found in loanwords.
Within the Germanic component, 'o' occurs as an allophone of short 'u' before the consonants 'r', 'kh' and 'khw'. This is part of a more general short vowel lowering rule that also lowers short 'i' to short 'e'.
German by comparison has a short 'o' phoneme which mostly developed from PG short 'u' and which is found in many contexts. Notable are occurences of 'o' derived from 'u' before nasal vowels, e. g. fromm, Sommer, Sonne, kommen, besonderer, gesponnen, geschwommen, genommen. (examples from Bin-Nun). This shift began in Old High German and continued through Middle High German and Early New High German.
The short vowel lowering rule survives in the German dialects but it is quite restricted geographically.
Yiddish occupies an intermediate position between Gothic and German. The short vowel lowering rule survives except in the Northeastern dialect (Litvish) which has lost the distinction between short and long vowels.
Words like vortsl, dorsht, shtorem, vorem, gorgl, vokher, etc. are universal in Yiddish and show that this rule operated at the earliest period in the history of the language.
On the other hand, there is well-established short 'o' phoneme in found in many word from the Germanic component such as groshn, holts, honik, shlos. gebot. etc. But 'u' before nasals was rarely lowered so that the listed German words above are represented in by Yiddish words that, historically, have short 'u' e. g. zumer, zun, kumen, etc.
I say 'historically' because in almost all the Yiddish dialects, the short 'u' has been transformed into another vowel. The excpeion is Alsation Yiddish. Alsation Yiddish is also exceptional in that while it has zumer, zun, kumen, etc. these are in free variation with forms that have 'o'
The picture that one gets is of an early Yiddish that had the short vowel lowering rule and lacked an 'o' phoneme but subsequently acquired it through contact with German, Hebrew and other languages that have it.
A particularly interesting set of words are fun, duner, and ful. The German cognates of these words have had 'o' since Old High Germanic times. It is possible that Yiddish preserves Old High German forms that are not attested in the literature. Alternately these forms may go back to an earlier Gemanic language such as Gothic or an earlier stage of the West Germanic language that developed into Old High German. Either way we are looking at a date for the origin of Yiddish that is earlier than 800 C E.