Vows As Oral Torah

במדרש ונשבעת חי ה' באמת כו'

In the Midrash on the verse "and you shall swear, 'As Hashem lives,' in truth, etc." (Yirmiyahu 4:2).

The Sfas Emes opens with the Midrash on the verse in Yirmiyahu that one must swear by Hashem's Name truthfully, setting up the discussion of vows and oaths.

כי כמו שהוא בפרט כן בכללות בנ"י זכו למצות נדרים ושבועות אחר כל המ' שנים של דור המדבר

For just as it is by the individual, so too with the entirety of Bnei Yisrael: they merited the mitzvos of nedarim (vows) and shevuos (oaths) only after all the forty years of the generation of the wilderness.

Just as an individual must first reach a certain spiritual level, so too Bnei Yisrael as a whole became fit for the mitzvos of vows and oaths only after the full forty years of refinement in the wilderness.

ונתקיים בהם ה' אלקיך תירא ואין להם עבודה אחרת רק התורה לכן נבלעו ד"ת בהם עד שהיוצא מפיהם חל עליו שם שמים

For in them was fulfilled "Hashem your God you shall fear" (Devarim 6:13), and they have no other avodah but the Torah; therefore the words of Torah became absorbed into them to such a degree that whatever issued from their mouths had the Name of Heaven rest upon it.

Because Bnei Yisrael fulfilled true yiras Shamayim and devoted themselves to nothing but Torah, the Torah became so absorbed within them that even their ordinary speech carried the sanctity of the Name of Heaven.

וזה ענין הנדרים ושבועות ובו תשבע והוא בחי' תורה שבע"פ לכן ניתן להם בסוף המ' שנים כשנכנסו לארץ ישראל:

And this is the matter of nedarim and shevuos -- "and by Him you shall swear" (Devarim 6:13) -- which is the aspect of Torah she'be'al peh (the Oral Torah); therefore it was given to them at the end of the forty years, when they entered Eretz Yisrael.

Vows and oaths, through which Hashem's Name comes to rest on a person's words, correspond to Torah she'be'al peh, and so they were given specifically at the end of the forty years when Bnei Yisrael entered Eretz Yisrael.

Summary: The Sfas Emes explains why the mitzvos of nedarim and shevuos were given to Bnei Yisrael only at the close of the forty years in the wilderness, as they were about to enter Eretz Yisrael. Through their fulfillment of true yiras Shamayim and their exclusive avodah of Torah, the words of Torah became so absorbed within them that the very speech issuing from their mouths could carry the Name of Heaven. This is the inner nature of vows and oaths: that Hashem's Name rests upon a person's own words. They therefore correspond to Torah she'be'al peh, the dimension of Torah that lives within a person, which is why they were granted specifically at that culminating moment of entering the Land.