Firefox's new extension API is Chrome's extension API + a few extra APIs. There's a handful of APIs that are actually Chrome/-ium specific, which made no sense porting to Firefox in the exact same way, but those should not be critical to the functioning of a password manager.

The only way in which Firefox is actually more restrictive with extensions, is in requiring user opt-in for internet connections that are not necessary for the functioning of the extension (so that's usually telemetry or ads).

But if LastPass refused to function because it can't send data back home that's not actually needed, and which it can't reasonably explain to users, or because it can't load third-party content from the internet, which would be a massive security hole, then I strongly advise using a different password manager.

Overall, though, this sounds to me like LastPass is just too incompetent to port their extension, which again should not be hard as Firefox's extension API is essentially a superset of Chrome's.