Many things in life are a negotiation and this is one of them.
- If you need the job desperately, fill out the paperwork and wait for the offer.
- If you are not desperate, I suggest the following
Ask for an offer letter. Tell them politely that you were wondering if they had made a decision yet as to whether or not they were going to extend you an offer for a position.
If they did not describe the position, as for the description of the position they offer to you. Even if they described this in the solicitation for the offer, ask for a clarification regarding the responsibilities, a list of direct reports (subordinates), and who would be your supervisor.
Ask for the salary, compensation (bonuses), and full description of benefits (e.g. heathcare, vacation and PTO days, holidays, pension / 401k). If there was a signing bonus, make sure this is in writing in the offer letter. Ask for the number of vacation days you will have when you start.
Ask for the expected start date or be prepared to supply a start date.
Ask for a decision date: when do they need to know your decision.
Consider their delivery of an employment contract / NDA / non-compete / request for drug test to be part of your decision process. Use that information to help you decide if you want to be their employee. Maybe their 'ask' is too onerous?
If they balk, ask what they need from you in order for them to provide an offer letter. It is fair for them to ask for references, background check, and perhaps even a drug test before making an offer. It is also fair for you to push back and tell them that they can provide you an offer letter than is contingent upon you meeting the terms of employment (e.g. they can give you an offer that is only good if you sign within xx days, pass a background check, pass a drug screening, ...).
I do not believe it is fair for a professional, salaried employee to be required to submit an employment contract to get an offer letter. You should not need to commit until they have completely described their offer to you.
The only exceptions I can think of are related to the government, military, and classified work.