Since LoginA is not member of sysadmin server role, the login must need IMPERSONATE permissions to use EXECUTE AS i.e.

USE dbB;
CREATE USER LoginA for login LoginA; -- This will create user under "public" database role
GRANT IMPERSONATE ON USER::LoginB to LoginA; -- This is required to use "EXECUTE AS"

Once LoginA got IMPERSONATE access, you can create procedure as follows in dbA where LoginA got full access (db_owner)

use dbA
go

CREATE PROCEDURE [UserAfterRestore] 
    @TargetDBName varchar(128),
    @GranteeLogin varchar(128),
    @ExectuteAs varchar(128)
AS
BEGIN
    SET NOCOUNT ON;
    Declare @TSQL varchar(2000);

    SELECT @TSQL = 'USE [' + @DBName + ']; 
                    Execute as user = '''+ @ExectuteAs +'''; 
                    IF NOT EXISTS (SELECT name FROM SYS.database_principals WHERE name = '''+ @GranteeLogin +''')
                        BEGIN
                            CREATE USER [' + @GranteeLogin + '] FOR LOGIN ['+ @GranteeLogin + '];
                        END
                        ALTER ROLE [db_datareader] ADD MEMBER ['+ @GranteeLogin +'];
                        ALTER ROLE [db_datawriter] ADD MEMBER ['+ @GranteeLogin +'];'
    Print 'Executing: ' + @TSQL

    EXEC (@TSQL);
END
GO

Call procedure as follows:

exec dbA.dbo.[UserAfterRestore] 
        @TargetDBName = dbB, 
        @GranteeLogin = 'LoginA', 
        @ExectuteAs = 'LoginB'; 

Since LoginA is not member of sysadmin server role, the login must need IMPERSONATE permissions to use EXECUTE AS i.e.

USE dbB;
CREATE USER LoginA for login LoginA; -- This will create user under "public" database role
GRANT IMPERSONATE ON USER::LoginB to LoginA; -- This is required to use "EXECUTE AS"

Once LoginA got IMPERSONATE access, you can create procedure as follows in dbA where LoginA got full access (db_owner)

use dbA
go

CREATE PROCEDURE [UserAfterRestore] 
    @TargetDBName varchar(128),
    @GranteeLogin varchar(128),
    @ExecuteAs varchar(128)
AS
BEGIN
    SET NOCOUNT ON;
    Declare @TSQL varchar(2000);

    SELECT @TSQL = 'USE [' + @DBName + ']; 
                    Execute as user = '''+ @ExecuteAs +'''; 
                    IF NOT EXISTS (SELECT name FROM SYS.database_principals WHERE name = '''+ @GranteeLogin +''')
                        BEGIN
                            CREATE USER [' + @GranteeLogin + '] FOR LOGIN ['+ @GranteeLogin + '];
                        END
                        ALTER ROLE [db_datareader] ADD MEMBER ['+ @GranteeLogin +'];
                        ALTER ROLE [db_datawriter] ADD MEMBER ['+ @GranteeLogin +'];'
    Print 'Executing: ' + @TSQL

    EXEC (@TSQL);
END
GO

Call procedure as follows:

exec dbA.dbo.[UserAfterRestore] 
        @TargetDBName = dbB, 
        @GranteeLogin = 'LoginA', 
        @ExecuteAs = 'LoginB'; 

If the target is multiple databases, then LoginA must be either member of sysadmin server role or must be user of all databases (under public role) and GRANTED for IMPERSONATE on respective database's db_owner