Mere permission to publish is no use if you want to make this software open-source (and I'm assuming you do, because otherwise this question's off-topic!).
The important thing is to get a clear statement of the necessary permissions from the rightsholders. You could list those permissions out (eg, I need to be able to share the code with others; I need you to license any patents you hold that are embodied in the code for everyone's use; I need to be able to host the code on this server, or that service; I need to be able to require others who re-share it to do so on the same terms). But experience with crayon licences suggests that it doesn't work well when random users without legal training try to codify legal requirements.
So the easiest and clearest way to do this is to get them to give it to you under the same licence under which you intend to distribute it to others. Standard free software licences are written in well-tested legal verbiage, and are suitable for corporate legal departments to read; they'll know where they stand if they are asked to OK this. Moreover, if they have problems with giving it to you under, say, GPLv3, then it's reasonable to assume they'll not be happy with you sharing it with others on those same terms.
So decide on your licence, give them a copy of that licence, and ask them to give you a copy of the source code under those terms. Then you'll know exactly what you can and can't do with it.