If a girl does not have enough properly Tznius clothing and is therefore dressing not Tznius, is she allowed to buy Tznius clothing during Sefirah?
There is NO dispensation ever for wearing non-Tznius clothing.
On the other hand, not wearing new clothing during Sefirah is merely a custom and is waived where it is done for the purpose of a Mitzvah, especially such an important Mitzvah which involves also many potential Issurim.
True, it is a Minhag (not mentioned in Shulchan Aruch) with sources in early Poskim (including the Leket Yosher – in the Minhagim of the Baal Terumas Hadeshen and Teshuvos Upsokim Chachmei Ashkenaz – Siman 69) to not buy clothing during Sefiras Haomer (except for Lag B’omer). However, in any pressing situation (i.e. where one will have a financial loss or they will be uncomfortable until new clothing are bought) and one can not wait until Shavuos, one may buy clothing during Sefirah (however it is best if it is first worn on Shabbos or Lag B’omer). In this case, there is no reason to wait and new clothing should be purchased immediately
All this applies to clothing that are considered ‘Choshuv’. However, regarding clothing that are insignificant, there is more room to be lenient.