Is there any way so IIS could connect to LocalDB without using the NT SERVICE\NETWORK SERVICE
user account.
This account has not suitable permissions. I'm looking use some other default account or is there some way that I can use the NETWORK SERVICE
account without changing permissions?
You should use Shared Instances feature of LocalDB. These two posts on Using LocalDB with Full IIS should give you more information. Especially the second part seems relevant, but the first one contains some context as well.
(note: the original links are no longer available, using archive.org instead)
Original (non-working as of March 2019) links: