Yes LastPass can Autofill when logging in to apps on your iPhone, as long as LastPass is set up for Autofill per these steps: https://support.logmeininc.com/lastpass/help/how-do-i-enable-and-use-autofill-in-the-lastpass-passwo...
Please note that due to a limitation in iOS, LastPass isn't able to automatically prompt to save new logins when you log in to an app, but you can add these sites to your Vault in the LastPass app. To do this, Open the LastPass app, tap Add in the upper right and choose what kind of entry you want to add to your Vault. LastPass will suggest several websites, or you can type the site name or URL to add it to your Vault.
A belated thank you. That worked well.
I have a similar problem on Win Desktop with apps that are not opened from within a browser.
How do I get LP to automatically prompt my login/password
We would recommend adding the URL for the site associated with the app in the URL field when adding an entry/item to your Vault. It is up to the App/website creator to make sure that their App is associated with a URL so that when adding an item to a password manager, it will be able to use the URL for the site to autofill correctly in to the corresponding app. For example, for the Twitter app you would want to put https://twitter.com in the URL field.
Some apps, we have noticed, are not correctly associated with a URL and you may need to add the app name to the Notes field in your Vault entry for that app. So for the previous example, in the Twitter.com vault entry if the app is not correctly showing you the Twitter.com entry in your Vault when logging in you would add "Twitter" in the Notes area for that vault entry. This way, when you are logging in to the Twitter app if the twitter.com entry isn't automatically shown to log in with you can search "Twitter" and LastPass should show you the twitter.com Vault entry.