Your options for borrowing money for moving expenses are really the same as borrowing money for anything else:
- Take a loan out with a bank
- Open a new credit card and hope you can get a high enough limit
- Borrow from friends or family
- Not borrowing, per se - sell a bunch of stuff or do more DIY/recruit friends or family to assist.
Obviously, there are pros/cons and caveats to each option and your options will be limited by your credit score and income factors.
The best approach, in my opinion, is to discuss this with your employer. Let them know that you don't have the funds available to pay for moving expenses upfront and be reimbursed for them later. I would ask if they could put the major expenses onto a company card instead. This will likely result in a more scrutinized expense review (finding a cheaper mover or van rental company for instance), but that's just the compromise. That might not be the case and they might just let you charge what you feel is appropriate, but just be aware of that.
There is a potential benefit in opening a new credit card. It'll be hard to get approved for anything with a sub-30% interest rate or a higher card limit because of your score, but there are options out there. If you get the card, only use it for moving expenses and subsequently pay it in full with the reimbursement and then don't use it again, there is a benefit to your credit score in a longer view of having more available credit compared to credit utilized. Obviously, the major potential drawback is that you aren't disciplined and end up maxing that one out too resulting in even more financial problems. You score takes a small initial hit on the credit pull to open the account, but that goes away relatively quickly. This is not the best approach or one recommended lightly. This is just an alternative thought process for consideration.
Borrowing from friends or family is generally strongly discouraged because it rarely ends well. However, because this really is a short-term loan, I still feel that it's valid for consideration if it's a possibility. Again, not the best approach, and one I would not recommend for normal borrowing situations. But, because you would be reimbursed quickly (presumably), I see the risk as much lower for recommending this if you are a trustworthy person that will actually pay back the loan right away rather than risk a relationship by spending the reimbursement on other things.
To reiterate because this answer got longer than I originally expected and I want to make sure it sinks in: Don't borrow money unless it's the only option actually available. Go talk to your employer and try to figure out a better method for paying for these expenses.