Can a lower enlisted soldier recommend another soldier for an award for a mission that we had just finished?

I applaud your gumption in trying to recognize a fellow soldier. Short answer is - yes you may recommend said soldier for an award. Follow the guidance provided in AR 600-8-22 and submit IAW your local policies. That said, were your DA Form 638 to cross my desk, you had better be tight in your writing and capture the facts of the matter well. I would have questions of why someone with more experience did not see the said actions or submit an award for said actions as they should have keener insight into what truly is exceptional performance for a soldier in that position. Good Luck in your endeavor and thanks again for taking care of soldiers

Yes, you can write/submit the award. The regulations state that the submitter should be of higher rank and/or someone with direct personal knowledge of the situation being recommended.

If you do, you should seriously ensure your writing is up to snuff. Have those that are seasoned in award writing proof-read your stuff, regardless of your proficiency.

Personally, I think it's a great idea you wanting to do this.

3–5. Who may recommend
a. The Army does not allow self-recognition, meaning that a Soldier may not recommend themselves for award of a decoration. Recommendations by spouses and other Family member are also considered
self-recognition.
b. The recommender must be senior in rank and associated with the recommended individual by virtue of their position within the chain of command. Additionally, the recommender must have knowledge of all the actions or service cited. That is, the recommender must have either observed the actions or been provided information by an individual who observed the actions.
c. If the recommender is not the commanding officer, the commander, if available, must endorse the recommendation. If it is no longer possible to route the recommendation through the commander (for example, the commander is deceased), a signed statement to the effect must be included. In this case, another officer who has knowledge of the actions and who was senior in the chain of command of the individual being recommended during the period for which recognition is desired may endorse the recommendation.
d. Recommendations for the award of Army decorations to members of another Service branch will be submitted by Army personnel only.

AR 600–8–22 • 19 January 2024 47

Basically the same as what was in the 2019 version which was the last version before this one.