Routinely asking your clients to gauge their experience should be a common practice for any business.  Repeat business says a lot but fine tuning the ways you interact and conduct your services is the best way to make improvements.  Forming a solid questionnaire to send to clients after a job is completed can show in detail the areas your company can work on in the future.  Your business might work with vendors as well and a few tweaks in the questions can help your relationships run smoother.  The simple fact that you are asking for feedback is a way of showing you want to establish a lasting relationship with these parties.
When formulating a questionnaire, circulate ideas between everyone in your company that is involved.  You want to understand as many of the issues and concerns that can arise.  Having a one-time meeting is not sufficient.  Brainstorming is a good idea to get the creative juices going but people will always think of other and better ideas the minute they leave.  I recommend having an initial meeting then opening an online document that many can add to.  Run through the practices from beginning to end and add whatever concerns come up later.
There are some key points every business performance questionnaire should ask their clients including but not limited to:
Communication –  “Did our representatives respond to you the client in a timely, respectful, and appropriate manner?”
Timeline – “Did the project flow according to the established timeline?”
Budget – “Were there concerns about how the budget was spent?”
Professionalism – “Are you pleased with the quality and equipment used in our services?”
Completeness – “Did we assist you through the entire process you hired us to perform?”
Product Satisfaction – “How satisfied are you with the product/services we provided?”
Effectiveness – “How well did our services improve your business?”
Open Ended – “Are there any ways you would like to see our performances improved in the future?”  Always provide an open-ended write-in question as you can no client completely fits into a box and they might have something specific to add that was somehow glossed over or omitted.
Relating to video production, you can be more detailed with the questions.  Asking about each aspect of the video such as:
Script – “Were you pleased with the creative script ideas we presented?”
Visual Quality – “Were you pleased with the resolution output and overall design?”
Actors – “Were you pleased with the look and quality of those we chose to be on screen?”
Music and Sound – “Were you pleased with the musical choices?”
Text and Graphics – “Were we able to provide appealing and appropriately annotated graphics to the video?”
Editing – “Were the choices in editing cuts to your liking?”
Pace – “Did the video lag in engagement at any points?”
Viewership – “Did the video reach your expected viewer rate?”
Distribution – “Did we provide accurate and sufficient methods of distributing the video to your intended viewers?”
Many of these questions can be further broken down into smaller categories and you can add or subtract ones that might apply to each project.  You want to cover all the aspects that you represent.  You know your company best so take input from everyone you can.  As you continue in this process you will find newer ideas to add to the questionnaire from the open-ended responses.
If you can make the effort to assign most of these questions as Yes or No answers or on a 1-10 grading scale (aside from the open-ended write-in response) then it will be easier for the client to fill out, given their busy schedule and more possible for you to gauge the areas your company needs for improvement.  A numerical scale can is the most accurate way to monitor those troubled areas.
Getting the wording right is crucial as the way the question is phrased will derive a different response.  If the questions are Yes or No then word them as such.  Don’t leave things confusing.  Also, try to phrase them in a positive manner such as “Were you pleased with the script ideas?” as opposed to a negative sounding question “Was there anything wrong with the script?”  Your questions should be specific and goal-orientated in the focus of improvement rather than scoring well.
The best time to submit your questionnaire to a client is about a month after the project has been distributed.  You want them to have sufficient time to assess its effectiveness but you do not to wait so long that they forget the details of working with you.  Know that you will not get a response from everyone you send one to.  Hopefully, your results accurately reflect the overall general experience of your clients rather than just the angry ones.  Offering the questionnaire in a digital format is a far superior method as it will save time and you will receive more responses.  You should have a good enough relationship with the client company that you can send it directly to the appropriate person, this should, of course, be the person who you were most in contact with but might also be someone higher up who will make the decision to reuse your services or not.
After a client completes the questionnaire, send them a thank you email.  Let them know their responses are greatly appreciated and it will help for future business engagement.  You do not need to defend each negative response and you should remain professional about the feedback.  Handling negative criticism is part of the game so make it constructive.
Once you gather a sum of questionnaires it is up to your team to evaluate and apply the feedback.  It is best not to assign blame for certain issues based on a single project.  Learn what you can and give issues time to improve on their own then discuss what issues are most pertinent and govern how to best approach them.  If there were issues on a specific project then add notes to the feedback about why this problem may have happened.  In the end, you will find this is a rewarding and insightful solution to growing your business.