The important term you want to know is 'Modal.' A modal form or dialog is one that is going to cause it's parent to stop running until its own process is finished (sorry if you knew that already).
I am pretty confident that that VBA msgbox is always modal. There's no way to put one on the screen and have the macro that created it continue running.
However, Userforms can be non-modal. You can set up a userform to look a lot like a msgbox, if you want.
There may be better solutions, but this is how I'd do it. Remeber that you'll have to figure out when you want the userform to go away and either hide or unload it as necessary.