When applying to an opportunity, almost all candidates need to submit a resume. Your resume should include facts and figures that highlight your previous responsibilities and accomplishments. Your cover letter is is a supplemental marketing tool to provide additional context and personality to your resume and ultimately convey why you're the right person for the job.
If the goal of a cover letter is to put your best foot forward and stand out, I would refrain from highlighting terminations or past disciplinary action you've faced.
"But if I choose not to disclose my termination on a cover letter, isn't that being dishonest?"
No. Your initial application, resume and cover letter, is your opportunity to highlight why you are the most qualified candidate. If a Hiring Manager or Recruiter read your documents, and you piqued their interest, they'll invite you to connect on a phone or video interview. They'll want to learn more about you and may ask why you left your previous positions.
"Great, so I'll speak to a Hiring Manager on a call, and what happens when they ask me live in-person or on a call? Isn't that even more awkward?"
I believe in telling the truth no matter what, especially on an interview. However, there are ways to answer a question honestly, without damning yourself in the process. Provide a thoughtful statement regarding why you're no longer employed with that organization and then highlight what you learned with that company. Remember, you're not the first person terminated and you won't be the last. If you articulate what you learned from that company and how you turned a negative situation into a positive one, you're far more likely to proceed and find success during the interview process.