Here's the scenario:
- Friend started a new position (work in a medical lab) within his company about 3 months ago
- He's the only person working in this department (prior person was fired)
- He's straightened out the mess that was left by the prior person
- He was told "you can't take an extended vacation until we find someone"
- The problem is they're even posting or looking

Could he tell his boss that he wants to take a week off in January so he'd like to train someone for while he's gone?

It doesn't seem fair that they leave him in limbo like this.

Thanks in advance for your help

He can ask his boss but his boss does not have to agree. It is up to the employer/boss when vacation can & cannot be taken.

He's only held the position for three months and he's concerned about taking vacation?

In this position, it's been 3 months. He was working in a different lab before. After starting at this lab, he was been putting in 60-70 hours per week to get caught up and straighten things out. To keep up with the current workload, he's working 50-60 hours a week. Since he's salaried, he doesn't even get overtime for all his effort.

I realize the employer has to approve the time off but the employer promised (verbally) to provide additional help. Can anything be done to nudge the process along?

Should have said he's worked for this company longer. It's been 3 months since he moved to the new area.

It's still up to the employer/boss as to when he can & cannot take vacation. There is really not much he can do to get his employer to hire additional help.

This may seem unfair but it is not illegal.

Since he's salaried, he doesn't even get overtime for all his effort.

Just to be clear, "salaried" is just a payment method which does not mean much by itself. Exempt Salaried employees have no right to paid overtime, while Non-Exempt Salaried employees do have a right to paid overtime. It is not the "Salaried" method that takes away paid overtime but rather the Exempt status (assuming the employee is actually Exempt).