Can my CO just decide to never sign an award I submitted?

Ma'am, in AR 600-8-22, Military Awards, dated 5 March 2019, section 3-21 will have your answer. But, I want to bring two subsections to your attention:

j. Each headquarters and organization will establish internal procedures to ensure that every award recommendation is processed with a minimum of delay. Recommendations requiring action by HRC or higher authority should arrive no later than 60 days prior to the desired presentation date. All other recommendations should be approved or disapproved within command channels within 60 days of initiation.

and

n. Once the recommendation is entered into official channels, commanders at each echelon have 10 calendar days to take action on the award commensurate with their authority. The award must continue to process through the appropriate channels to the approval/disapproval authority.

Now, I also want to bring to your attention that there is no regulatory guidance/requirement to approve/give a Soldier an ETS nor PCS award. Are/were the Soldiers flagged? Have you asked why there has been no movement? Also...if memory serves, the awards need to be processed thru ePAT, which involves S1.

Oh...one more thing. As for the "anyone can submit," that is technically correct but also technically incorrect. The regs state the submitter should be a person that has direct personal knowledge or superior in rank. What this means is that...say your Supply Specialist wants to put in an award for the Supply Sergeant but doesn't know how or that they think the award will go nowhere because of their rank. Now, while the Specialist has direct knowledge, they could come to you and provide all pertinent details to you and you can then write up/submit the award.

This crap happens all the time.
Intermediate authorities may recommend approval, upgrade, downgrade, or disapprove. That's it, no matter what they have to check one of 4 boxes and send it to the next level. Only the true approval authority can decide yay or nay.
Our Officer Corps needs to do better with awards, stop worrying about "looking bad for recommending downgrade" and just sign the damn form.

If it was me, I'd do a little more than email, I'd get on the phone and respectfully ask the CDR if he/she had received the the award request and what was the status. You are the XO.

MSG (Join to see) has a great answer. If you're the Executive Officer for the unit, you should be able to go directly to the commander and ask about the awards. Working with your S1, you should review the awards process to ensure it is in compliance with Directives and policies. Also, you should look for unnecessary approvals and reviews in the process. You may be able to eliminate those. There should be a unit awards officer who has the experience and responsibility to review all award submissions. In some cases, there may be an awards board that reviews submissions and makes a recommendation to the Commander. You should find a book on Lean process improvement and read about process mapping. It will be useful in understanding and improving the awards process.

The commander has the option to sign it, down grade it, or send it back as unsuitable. In a nut shell, they don't have to sign any award the don't beleive is mertited

I retired in 1993. In 2002 I received an award from my last unit. Nice of them but though it was signed and approved before my retirement but no one awarded it to me. When the unit rotated to Ft Benning from Fort Knox a clerk found it in the files. He forwarded it to me at my HOR.
Mistakes happen. Odd - No one at the diner that afternoon was impressed enough to buy me a coffee, but my mother liked the colors.

Is this about you wanting the cdr to sign your recommendation for an award or the soldier not being recognized for service you deem commendable . Being in a unit for 13 years is not an accomplishment what they accomplished and how they served is what warrants the award.

Yes.

What was the award for, it may be the approving authority is the Battalion commander if the award was high enough.