You have to work much more closely with your vendor, this means the party that insists that this need to be a hourly fee project is thinking that there are unknowns in the project or don't understand the project fully yet,
I) - The most important thing is you need to clearly articulate what needs to be done as listed below,
a) - Provide a well documented clear thorough high quality FUNCTIONAL SPECIFICATION with every single detail called out, include drawings, screen layouts, pictures, examples of complicated scenarios, provide as much details as you can - I would have this ready as early as possible and even bid out the project with the specification after getting the NDA - The deliverable you are getting is going to be of the same quality of the functional spec you are giving - so make sure the functional spec is of really high quality
b) - Provide a well documented TEST SCRIPT with all possible scenarios that you are going to test for with thorough examples
c) - Provide all the TOOLS and SYSTEM ACCESS that are needed in order to execute this project successfully, access to all the systems, get them any open source tools or third party tools,
d) - Clearly call out the LIST OF DELIVERABLES and agree on DATES, that these MILESTONES need to be hit
II) - You have to have at least weekly calls with your vendor, if not may be once in two/three days or in worst case scenarios do it daily
III) - Any question that comes up address it immediately
IV) - Over communicate, not just telling them what needs to be done, look for signs that they have understood what you are asking - that is why you always need to do video conference meetings - so you can read the body language - many people will shake their head or say yes - but if you see their body language/read their face you will infer they don't completely understand what you said - so you better explain with pictures and diagrams clearly - make sure they understand everything thoroughly - If they don't understand something tell them to stop the work and call you immediately
V) - If you keep doing this it gets better, make some small wins, hit a couple of milestones until everybody feels good about the results
VI) - But I would continue to do this until you guys make a big delivery
VII) - I would also provide constant one on one feedback to the sponsor on the vendor side, if you think certain resources need to up their game, or you need the resources switched based on their performance
Once you are happy that they are performing to your expectations, you have confidence trust built on each other then you can may be switch to a fixed budget project or if the other party is not performing, missing milestones and dates, you can pull the plug on the project.
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