You are giving very little to go by. Pretty unlikely that this is caused by a problem in your program, far more likely that this is an environmental problem that's specific to the user's machine.
Some background. The TextBox control in a Winforms app is the exact same component that Notepad uses to allow you to edit text. The underlying native component is an Edit control, it has been a standard component in Windows since version 1.0. Note how the context menu you get when you right-click the TextBox and Notepad is identical. There is no distinction between the Paste command in that menu and pressing Ctrl+V, they both cause the WM_PASTE message to be sent to the native Edit control. Which internally deals with the clipboard, that code is entirely outside of your reach and operates identically in Notepad as it does in your program.
A thread apartment problem is unlikely, the customer ought to also have a problem with the Copy command and you should have noticed it before. In C#, it is set by the [STAThread] attribute on the Main() method, it is compiler-generated in VB.NET.
There are plenty of utilities that can cause the clipboard to misbehave. First and foremost are clipboard viewers, programs that hook into clipboard notifications. AddClipboardFormatListener() is the underlying winapi function. They tend to do things like "enhancing" the clipboard by allowing it to store more than one item or give another view of what is on the clipboard. They tend to destabilize the machine by not passing notifications on to the next viewer properly or by breaking the viewer chain by not unregistering themselves correctly. Such a broken chain in itself is a lead to "works okay in Notepad, not in mine".
Such kind of problems are very hard to diagnose of course and don't typically get resolved until the computer user thoroughly cleans his machine. That this is his problem and not yours is always difficult news to deliver, can't really help you with that.