Could the House/Senate assign committee assignments without political parties?

[True Federalist (진정한 연방 주의자)]

Short answer:
Yes, but they won't.

Long answer:
The House and Senate can devise whatever rules a a majority of that house chooses to use. The Constitution doesn't require political parties, was written at a time when there were no national political parties, and was written by people who by and large considered parties destructive to good government.

However, once parties develop, they generally are used as the means of organizing the legislature and neither party stands to benefit from making things easy for independents to stand aloof from the fray, so while non-partisan allocation of committee assignments could happen, it won't.  At best, if there were a significant third party, they might be given a number of committee slots to allocate as they desire.