Keep it short and sweet. More than a short paragraph or two and you will loose them. If that means you have to craft personal mail for each one, take the time (I need X from you... Y from you... Z from you...).
- the huge impact they could have with as much motivational language as you can manage
- the scope of their involvement as specifically and accurately as possible
- Add a short one liner of why they are the perfect person and/or why this fits with their existing role
- I'm writing you today to ask for your help. The information you can give me will be a key ingredient in making my project a success. I'll be glad to share (some product of our output) with you if you are interested.
- If you can help, I'd need a 1 hour meeting, 4-5 hours of thoughtful contribution total over the next 2-3 weeks, and a follow up meeting to make sure we are on the same page. I want to wrap up the project by the end of next month.
- Thank you so much - having your extensive expertise helps me make sure my project meets the needs of your organization or people like you.
It's hard to say no to that.
But one caveat - I picked the < 1 day of total time above as an example of when it's pretty easy to ask for and get help from people who don't have a managerial directive to help you. Most professionals are glad to help others on the pay-it-forward/good karma model, as long as they see that they can be helpful and have a profound impact with limited commitment.
However, if you are shaking your head and thinking "but bethlakshmi, I need more than that!!!" then what you have is a big enough project that you need managerial commitment first. Sorry, but that's how it is. You can try this, and see if it works - you certainly won't offend anyone with a nice email. But if you need more than a day (as a general rule of thumb for Western culture/knowledge work) then you probably exceed a point where people feel comfortable giving you time, and need to reconsider what you are asking for and whether it's reasonable.
Lastly, any time you can get closer to a trade, you can probably subtly request more time/help - if you can give them your research, a sample product, a favor later (I'll owe you one!), then it's more of a bargain and less of a donation - which shows you put value on their time, and offers the idea that they will benefit too.