Get an appraisal first.
In the appraisal world, there's a rule: Whoever orders the appraisal becomes the appraiser's client. This is important because unless you are the one to order the appraisal, you will not have access to it once it's completed, even if you end up paying for it.
Imagine going to your bank or mortgage broker and being approved for a mortgage pending an appraisal. They call us to appraise your property, with you footing the bill. Once the appraisal is complete, which could take up to a week, the appraiser is required by law to send it directly to the bank. They are unable to share or discuss it with you without the bank's permission.
What happens if you decide to go to another lender? Well, the process starts all over again and you may have to pay for a completely different appraisal that you also do not have access to. In any negotiation, the one with the best information is in a position of power. If the bank orders the appraisal, you will be paying to arm them with that information.
For this reason, you should get an appraisal first - before going to your lender or broker.