Could Kamala Harris use “will” instead of “are going to” in this short viral video?

Solution 1:

We did it. We did it, Joe. You're going to be the next president of the United States.

Using "going to" suggests something that is imminent, an event that is about to happen in the not too distant future. Harris based her prediction on present evidence; not on a hunch, theory, or hope but on fact: the number of Electoral College votes won by Biden's team.

Saying that, she could have easily said: “We did it. We did it, Joe. You'll be the next president of the United States” and the meaning would not have changed.
Some grammar books explain that "will" is used for predictions, promises, facts and for instant decisions. The news of the victory means Joe Biden becoming the 46th President of the US is the promised or factual outcome i.e., Joe Biden will enter the White House on January 20, 2021.

In keeping with tradition, the inauguration will be held on Wednesday January 20, in front of the US Capitol in Washington DC.
(The Telegraph)

Relax. Biden will be sworn in Jan. 20.
President-elect Joe Biden will be inaugurated at noon on Jan. 20.
(The Washington Post)

As the inauguration date gets nearer, native speakers will probably say, “Biden is going to be inaugurated on Wednesday”.

It worth pointing out that the contracted form “'ll” is far more natural in casual speech whereas the full form is preferable in formal speech and writing.

Solution 2:

Both could have been used by Kamala Harris, but, "be going to" sounds more appropriate here.

"be going to" is about an action in a more or less near future. It is about something which has been planned. What's more, "be going to" sounds more enthusiastic, more energetic.

"will" is about the future as well, but it can be a distant or a near future. "will" adds the idea of willingness, of determination, something that you really want (!) to do.

Solution 3:

Both can be used, as shown here: https://www.grammar.cl/Notes/Future_Will_vs_Going.htm.

Going to is used when something has shown definite signs of happening.

Example: It is going to rain this evening.

And can also be used when you are definitely intending to do something.

Example: I am going to go to the park this evening.

Also when something is about to happen: His bag is going to rip, it's so full of books!

Will is used when you are predicting something.

Example: It will rain this evening.

And for rapid decisions:

I will go to the park this evening (I've just decided to go to the park)

And for making offers, threats, etc:

I'll give you the mango for two (currency)s today. It's a discount.