How could I do it without a database lookup for one form?

If you don't want to have a database, that means you'll need either formulas or Javascript to make it work.  And any changes to your list of departments and department heads will necessitate updating those formulas or Javascript.

Let's say you have three teams in your dropdown, Team A, Team B, and Team C, and for simplicity sake, let's say their email addresses are a@email.com, b@email.com, and c@email.com respectively.

If your department drop-down field has a variable name of select_department, then you could populate the email addresses into a single line field via a variable with nested IF statements.  This is tested on both the Classic Designer and the New Designer on Version 11.0.2212.30907:

=IF(select_department="Team A", "a@email.com",  IF(select_department="Team B", "b@email.com", IF(select_department="Team C", "c@email.com", "")))

Of course, that may get fairly unmanagable if you have a lot of departments.

You could also do it via Javascript.  Using the same examples as above.  Give the Department dropdown the CSS Class of selectDepartment, and give the Department Head Email field the CSS Class of departmentHeadEmail.

Here's Javascript tested on Classic Designer Version 11.0.2212.30907: 

//The document ready function is automatically called when the form loads.
$(document).ready(function() {
  
  //When the selectDepartment drop-down (select) field is changed,
  //the value of the departmentHeadEmail (input) field will be populated.
  $('.selectDepartment select').change(function() {
    var emailAddress = '';
    if($(this).val() == 'Team A') 
    { 
      emailAddress = 'a@email.com'; 
    }
    else if($(this).val() == 'Team B') 
    { 
      emailAddress = 'b@email.com'; 
    }
    else if($(this).val() == 'Team C') 
    { 
      emailAddress = 'c@email.com'; 
    }
    $('.departmentHeadEmail input').val(emailAddress);
  });
  
});

Here's Javascript tested on New Designer Version 11.0.2212.30907: 

//Reference the form field by class name - the [0] refers to the first 
//field with that class name.
var departmentField = LFForm.findFieldsByClassName('selectDepartment')[0];
var emailField = LFForm.findFieldsByClassName('departmentHeadEmail')[0];

//When the departmentField is changed, use it's value to determine and
//populate the value of the emailField.
LFForm.onFieldChange(function() { 
  var emailAddress = '';
  var departmentName = LFForm.getFieldValues(departmentField);
    if(departmentName == 'Team A') 
    { 
      emailAddress = 'a@email.com'; 
    }
    else if(departmentName == 'Team B') 
    { 
      emailAddress = 'b@email.com'; 
    }
    else if(departmentName == 'Team C') 
    { 
      emailAddress = 'c@email.com'; 
    }
  LFForm.setFieldValues(emailField, emailAddress);
}, departmentField );

Could I have multiple email addresses assigned to the same choice?  We have a department with co-department heads and I would want the form be assigned to both of them.

Would something like this work in your first example?

=IF(select_department="Team A", "a@email.com", "b@email.com")

This: 

=IF(select_department="Team A", "a@email.com", "b@email.com")

Is saying "if select_department is Team A, then return a@email.com, otherwise return b@email.com" so that wouldn't work.  

If you are familiar at all with how the IF formula works in spreadsheets like Excel, it's the same method here:   =IF(check if something is true, return this value if true, return this value if false).    In the example I gave before of three nested IF statements, instead of "return this value if false", we went into a second IF statement, and if that was false, went into a third IF statement.  It was only if all three were false that if returned the blank string "" at the end.

In order to return multiple email addresses for the team, we need to make sure those email addresses are in the same value, something like this: 

=IF(select_department="Team A", "a@email.com,b@email.com", "")

That is saying "if select_department is Team A, then return 'a@email.com,b@email.com', otherwise return blank".  It's returning both email addresses within a single value, separated by a comma.

Now, the part I have not tested, is taking the variable from that email field, and plugging it into the "Email Service Task" in your process diagram.  I'm pretty sure it'll work, but since I haven't tested it, I can't say with certainty that it will like the comma between the two email addresses ('a@email.com,b@email.com'), maybe it would actually prefer a semi-colon instead ('a@email.com;b@email.com'), you'll just need to test that part to see.

I'm actually trying to plug it into a user service task.  The employee submits the form with department head approval needed and I want it to be sent to the right DH so they can fill out their part of the form.  I would want it so that the employee can choose their department and the department head's email would be attached somehow so they can do the next step of the process.

So I feel like we're getting much closer, but I'm still running into issues.  I want to do the second option - one team for all the department heads separated into roles - but the code given doesn't seem to be working.  Here are the screenshots that I have.

You need to have them as a Team Member to have tasks assigned to them in that team.

Team Administrator lets them create/edit/delete the team, along with view and reassign tasks, and adding/removing users to the team - so you usually want to really limit who is marked that way.

Team Manager can view and reassign tasks, and adding/removing users to the team - so less restricted than Team Administrator, but you probably still want it to be limited to specific individuals.

And then Team Member is the actual members of the team.

And you can have more than one selected, so the Team Administrator for example could be also the Team Manager and Team Member.

Even though these are Department Heads, I'm guessing most of them are just going to be Team Members on this team of Department Heads.

That's the only issue I'm seeing on that screenshot, so hopefully that works.