Can one use a hyphen to form 2 words with same prefix
Solution 1:
In some cases of dangling hyphens, the hyphen is put (back) into a word that used to commonly have it or occasionally still does. For example "inter- and intra-species". You may be able to do something similar in this case: "neuro-degenerative and -inflammatory".
If you couldn't get away with the hyphenated forms, this wouldn't work. A good internal peer review is of course necessary in grant proposals - hopefully you have someone who would nitpick things like this.
Solution 2:
This is called "suspensive hyphenation," according to my copy of the AP Style manual.
However, I don't think you can do it here.
To start of with, most would agree your suggestion does not carry its intended meaning.
[...] the development of neuro-degenerative and inflammatory disorders.
This clearly discusses two types of diseases: neurodegenerative ones and general inflammatory ones.
Edd's example
"the development of neurodegenerative and -inflammatory disorders"
is better, but note that "neurodegenerative" and "neuroinflammatory" are single words, not hyphenated, see Mitochondrial Dynamics in Cancer and Neurodegenerative and Neuroinflammatory Diseases, for example, so this option makes little sense.
APA Style has a few suggestions for hyphen usage, here is "General Principle 5" (emphasis added):
When two or more compound modifiers have a common base, this base is sometimes omitted in all except the last modifier, but the hyphens are retained.
Long- and short-term memory
2-, 3-, and 10-min trials
Although this is the inverse case (you have matching prefixes, not bases), APA notes you should retain hyphens, which you can't do if you never had one in the first place.
(AP Style gives the example of "He received a 10- to 20-year sentence in prison, also with alternating prefixes, not bases.)
Conclusion
Basically, you can't do this unless you already have hyphenated words, so this is tantamount to sticking hyphens between any other morphemes and calling it a day:
He disagreed, and -mantled the structure she had built.
Unless you're writing for The New Yorker, I'd skip this.
Not only that, but most examples have changing prefixes, not suffixes.
I would suggest this is probably because modifiers in English come before their head, so people are used to seeing constructions like "yellow and blue book," but not so much "yellow book and car," which describes a yellow book and a car, not a yellow book and a yellow car.
In short, I wouldn't do this, it's confusing. You don't want someone reviewing your grant proposal to have to stop and back up mentally to understand your sentence!
Plus, "neuroinflammatory" is the same amount of words as "inflammatory" anyway.