I suspect this is less a concern of grammar as it is marketing. It's worth pointing out that I'm sure on several situations the phrase "Let's make America great again" was used, as well as "I'm going to...". By using a shorter sentence as the main catchphrase, it's easier to adapt it to certain situations. By avoiding "let's", the phrase is more of a demand rather than a suggestion - it could be said that many voters in the target demographic would be more appealed to by this kind of phrasing. I can say it to someone who agrees with me and the "let's" part is implicit. Or, I can say it to someone who disagrees with me, and it carries the connotation "You are preventing America from being great again". Perhaps the strict character restrictions of various forms of social media also meant that keeping the catchphrase short was of utmost importance.

Two excellent answers. My spin on it is "Vote for me and Make America Great Again" It's all about how you wish to interpret it. The meaning is the same. Remember, this is marketing. It doesn't necessarily need to be connected with reality. No, I'm not making a political statement. I'm referring to marketing.

I think Zach Smith is right. Each year we see new phrases in our languages ( all the languages ) and in english i've seen some sentences are being cut. Some frist part or words are cut... some middle parts too. I think this is only a ruleless shortening method by only removing the first word. Like what sometimes happens in lyrics and poems, The removed word that you should guess what it was. It is logical especially when it's about election quotes. They have to have a little poetric power... ( this is all my personal opinion )

Apple slogan "Think different", AT&T slogan "Reach out and touch someone", Burger King "Have it your way", FTD "Say it with flowers". None of these have let's as they are personal messages i.e. not to the whole crowd/country but you personally. So it’s not about “Let US” but more about YOU making America great again. My opinion. Peace!

I'm late to this party, but what you say resonates with me. "Let's <do something>" is far more powerful and positive rhetorically and psychologically than "<Do something>" is. Someone who understood that, and who was interested in inclusivity and cooperation, would have certainly worded it that way.