VB 6 provides a Clipboard object that makes all of this extremely simple and convenient, but unfortunately that's not available from VBA.
If it were me, I'd go the API route. There's no reason to be scared of calling native APIs; the language provides you with the ability to do that for a reason.
However, a simpler alternative is to use the DataObject class, which is part of the Forms library. I would only recommend going this route if you are already using functionality from the Forms library in your app. Adding a reference to this library only to use the clipboard seems a bit silly.
For example, to place some text on the clipboard, you could use the following code:
Dim clipboard As MSForms.DataObject
Set clipboard = New MSForms.DataObject
clipboard.SetText "A string value"
clipboard.PutInClipboard
Or, to copy text from the clipboard into a string variable:
Dim clipboard As MSForms.DataObject
Dim strContents As String
Set clipboard = New MSForms.DataObject
clipboard.GetFromClipboard
strContents = clipboard.GetText