As you write Jews generally do not shave. This being said, I know of two approaches to kulot relevant to your question. R Yehuda Balsam conveniently writes them up here
In a situation when one has a situation in which he will lose money,
or possibly his job, R Moshe Feinstein writes (O.C. 4:102) that it is
permissible to shave. However, during the week in which Tish’a b’av
falls out, since it is halachikly assur m’drabanan to shave, one may
not do so under these circumstances.
dinonline expands on this
As R Moshe Feinstein explains elsewhere (Choshen Mishpat 1:93), this leniency
only applies if an actual loss would be incurred. If appearing
unshaven merely causes ridicule or embarrassment, the leniency does
not apply. Clearly, this will also be true of the discomfort a person
suffers from not shaving for a number of days.
A similar ruling was given by R Shlomo Zalman Auerbach (Maadanei
Shlomo) concerning the Omer period: It is permitted to shave for one’s
livelihood, but it is otherwise forbidden.
[However], “Kuntress Liknos Chochmah” cites R Yosef Shalom Elyashiv
that today it is forbidden to shave during the Omer even for purposes
of one’s livelihood, since being unshaven today does not cause a
person to look odd. This clearly depends on a person’s environment and
in practice if a person’s livelihood might be at stake the consensus
is to be lenient.
The second approach is from R Yosef Ber Soloveitchik (again from here)
In Nefesh Harav p. 191 he writes that shaving was permissible
during sefirah. He reasons that when the bnei yisrael accept a new
minhag, that minhag must be patterned after an existing halacha,
either mid’oraisah or mid’rabanan. An aveil observes three major steps
in his mourning: 1) the ‘shivah’ period, observed during the first
week, 2) the ‘shloshim’ which is the remaining thirty days, 3) the
remaining twelve months. Rav Soloveitchik assumed that the aveilus of
sefirah was patterned after the aveilus of the twelve months. During
this period, both shaving and haircuts are prohibited. However, the
gemarah (mo’ed kattan 22b) writes that if he becomes disheveled ‘ad
sheyig’ar bo chaveirav’-until his friends scold him, he may cut his
hair and shave.
Thus, one who normally shaves every day, and refrains from doing so
for a few days would be considered to have reached this point. Thus,
it is permissible for him to shave during this period (it is important
to note that Rav Soloveitchik himself did not trim his beard during
sefirah, because he did not consider that to be sufficiently unkempt
to apply this din.) R Soloveitchik’s opinion is obviously not the
mainstream shittah, but it has been accepted by some of his Talmidim.