Is it a problem to submit an incomplete rental application to a landlord or management company?

It really depends which paperwork you are missing and how strict the management company / landlord is with its requirements.  Many mgmt cos / landlords require a full application to be handed in and some will not accept packages that are not complete.  It is ultimately situational, but attempt to have as much of the required documents handed in as possible.

If you are missing an employment letter or tax return, it may be too important and could negatively impact your application.  However, if you are missing a former landlord letter or a copy of a social security card, it may not be as much of an issue.

If you know that you can eventually get the missing paperwork in the next few days, inform the mgmt co / landlord and ask if you can hand in the remaining documents later without jeopardizing your ability to get approved.