Convincing management of things like this usually involve two parts - the cost and the benefit.
What's the Benefit?
I'm not sure what's out there in productivity metrics, but there's definitely some great research on the problem of sitting for too long in terms of health.
Here's a few, but I just Googled for "sitting for prolonged periods of time" and found them.
Are a few, there's more but note as you read how many may just be quoting the same research.
The thing is while most people won't put it this way, the company really doesn't care if people die faster as long as they don't really die on the job. To the best of my knowledge, OSHA or other employee safety related organizations haven't classified standing/sitting desks as a must have for office safety, so it's not going to hit the world of corporate liability.
However - companies are getting more aware of and concerned with saving money on health insurance. Having more healthy employees with better weight, more fitness, less stress actually does reduce the cost of health insurance in the number of claims and the severity of claims filed against the policy. I don't know whether you can get at the numbers, but you can make a connection of "standing/sitting desk"->"fewer claims"->"cost savings"
What's the cost?
So, a lot of work space improvements have the multiplication problem. Getting an awesome ergonomic something for the one person with a doctors note is one thing. Getting the same ergonomic thing for everyone is much more expensive. I've looked myself (because I have a deep love of a particular keyboard tray) and found that some of this gear is disturbingly expensive. Multiply that out and it's going to get expensive if the company gets one for everyone.
The typical way to address that is to only give it to those with a strong need - thus, the doctor's note. At the very least, it limits the cost of the equipment to those willing to do the paperwork.
So... the other way to win this one is to do the research into the equipment and see if you can find an option that is both useful and cost efficient. One trick can be to look at total cost of ownership - it's not just the purchase price, it's the long term expense. If there's a great warranty, this long term expense goes down. If there's a record of high quality, you'll save money on not needing much support from the facilities crew. Know the whole picture so you can answer these questions - it's what management will be asking.
Have a plan
Have a pilot plan in mind. "Buy it just for me" is probably not a great way to go. @Jeanne Boyarsky has the ultimate no-cost pilot program - and that's one option. Another might be "buy it for 10% of the people on a floor". Let people rotate through the spiffy desks for a month each - done on a first come first serve volunteer basis. Collect feedback at the end of the month from both the person and their manager.
This might be something you have to volunteer to run - in addition to your regular job.
The trick is - you are asking for a pretty big investment, so if you want to see the change, someone is going to have to step up and do the work.