According to the 역사정보 historical information section for 팔꿈치 of the Naver 국어 사전(Korean-Korean dictionary), translated here:
The modern standard Korean '팔꿈치' appears in 18th century documents as 'ᄑᆞᆯ굼티' (pawl-gum-ti, where 'aw' is the romanisation of the arae-a ㆍ ). In the 19th century dictionary 한불자전 Hanbul Jajeon, mention is made of '발굼치 跟 (the hanja for 'heel')' and '굼치 跟 (the same hanja)', and the former is analysed as a compound word.
In the entry for 발꿈치, we find the following:
The older form '발굼치' appears in 16th century documents. It appears to be compound of 발 and 굼치, but a clear analysis of '굼치' is difficult. Even so, an example '팔굼치' (foot + 굼치) is found in the 19th century.
A comment from 2010 on the website for the Korean Language & Culture Institute at Ewha Woman's University states:
It is possible to guess the meaning by looking at the meaning of the words used in combination with '꿈치'. [...] Judging from these meanings, the suffix '꿈치' can be thought of as having something to do with 'a curved or bent part exposed to the outside of the body'.
Although I disagree with the analysis at the end that '앞꿈치' is not used, seeing that the several rows of joints (distal interphalangeal, proximal interphalangeal, metatarsalphalangeal) of the human forefoot can and do curve, this idea that 꿈치 and "curving/bending" makes good semantic sense, especially if linked with the verb 굽다/굽히다.
Cross-linguistically it is not strange for 'elbow' to be a compound of 'arm' and 'curve/bend', attested e.g. in some of the topolects of Chinese, and less transparently with Germanic (including English, where elbow = ell + bow) too.
There is also a bound lexeme in many Korean dialects, -꿉이 (e.g. 팔꿉이 in Gangwon and Hamgyeong Provinces). It is unclear whether this is cognate to the standard -꿈치, or whether this corresponds with the noun 굽 hoof (of a horse, goat, or other hoofed animal) / foot of a bowl, and whether these are all actually the same family of cognates.