The mobile lastpass client stores your password in the device's key locker which, in theory, should only be accessible after you enter appropriate credentials. In this case fingerprint.
Of course, there is a massive difference between security on Android to iOS (in general anyway) and I'm not familiar with the Android implementations other than to say that many Android devices have no hardware based locker which potentially seriously restricts the level of security you could expect.
On iOS, this has been well tested and found to be very secure - hence the ongoing battle between Apple and the FBI.
When unlocked, the vault is maintained in memory. So in theory, it would be possible to gain access. However, it uses protected memory which should not be accessible to any other application. As iOS has a more secure and restricted base, this again is likely to be more secure there than on Android. Though perhaps an Android expert can give a more definitive answer.
The downside of securing lastpass with a fingerprint is that the USA's border control have decided that you can be compelled to provide a fingerprint but cannot be compelled to provide a password. This has been verified in the US courts.
When I travel to the USA, I remove all cloud services except those I don't care about. Certainly I uninstall lastpass and add it back later. That doesn't just apply to the USA of course.
Using a fingerprint is not just theatre but it certainly does reduce security. Not likely to be enough to worry most people though. Of course, I only use LastPass for less critical passwords. The most sensitive are kept in a separate vault accessed using a different cross-platform tool. That doesn't have fingerprint integration and I don't keep the vault open anyway unlike LastPass which I often have open.