Neither ADDS nor Azure MFA can do this.
However, I do not see that as a service limitation. The concept you are describing sounds to me quite retro:
* User convenience would not be great (user instructions, behavior for different MFA methods, error reporting).
* From a security point of view, it is a risk as MFA credentials pass through the application and could be caught/used somewhere else.
I actually do not feel good anymore with letting users enter credentials into applications at all - and that is what is already happening with plain LDAP.
My strong recommendation is to have a talk with your 3rd party vendor and ask if it is possible / on the roadmap to use modern protocols (OAuth, OpenID Connect, SAML).
I know this can be hard sometimes ... but the line of arguments is pretty straight forward and any software vendor can hardly deny it.
Also modern protocols give you a Single Sign-On possibility - both a security and a user convenience improvement.