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I shipped my iMac to Vietnam as excess baggage in the original box.  At the time, Philippine Airlines had a flat $150 rate for excess, but I don't know about now.  Fortunately the box was beat up just enough.  When I got to TSN Airport, the customs guy asked me "New of used?" and accepted used, which of course was true.  I shipped whatever peripherals I could in the same box on the non-glass side.  The rest I shipped in another box along with other assorted junk items.   

As you seem to be an American, I would advise you check all peripherals to be sure they will work with 220-240V.  I would not ship a printer as different models seem to be sold in Vietnam, even if the same major brands like Epson and HP.  If you bring a US printer, you likely will have a hard time finding ink.

Take any removable batteries such as AA's out of things before you ship.  I had some minor items like a mouse fried by the scanners.  I don't think it would have happened with the batteries out.

I also carried a full cloned backup on a portable hard drive in my carry on, just in case there was a total SNAFU and I had to replace the computer.  Fortunately I didn't need it but it did give some assurance.

Upon googling iMacs, they appear to be quite slim.  The PC I'm looking to ship weighs about 50 pounds and its dimensions are 24'x9'x26'.  Throw in the monitor, mouse, and keyboard and it's a hefty parcel.  Would I be better off finding a third party shipping company?  If so, does anyone have recommendations?

I'm also super worried about damage or theft.

I have bought one PC from Japan  Model NEC  PC DA7700GAW It was 10kg, But there is no safer way then carry by hand, It was new, and I had no problem with check-in or check-out. I think you should bring with you and accessories put in your language or you can buy in Vietnam such an as Keyboard  Monitor.

An iMac looks slim but it's box is a little bigger because it must accommodate the angle between the screen and the stand.  Mine was a 2010 so its box may have been a little wider than the current box.  This brings up another issue.  Avoid all-in-ones whether PC or Mac.  Mine had a repair problem while in Vietnam and I had to use a taxi to bring it in because it couldn't be carried on a motorbike. 

24x9x26 should still fit in the maximum size for international excess baggage.  Check on the airline website.  Personally, I would bring the PC, along with evidence that it is used for customs, and buy my monitor in Vietnam (220V) even if it costs a little more than in the US.  Even newer monitors are still quite heavy and of course you risk breakage and the hassle of making a claim.  Remember that a desktop PC will probably need a power converter as well although most laptops are dual currency because the charger is acting as an AC/DC converter. 

Another advantage of the airlines and bringing it with you as excess baggage has to do with your concern over damage and theft.  If something does go wrong, you will be dealing with an international company with a reputation to uphold as well as employees who speak English.

I would disassemble the PC into pieces and check them in. Make you to encrypt your harddrive though and if possible handcarry a backup or leave one with friends or so.
Or ask a friend to open a protected fileserver, so you can download it from vietnam.

You should definitely remove the cpu cooler before shipping. Most of them are heavy and you risk to dmg the cpu or the motherboard.

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Carrying with you better because simpler, safer, probably cheaper, have it sooner.
I would remove and hand carry the expensive bits: cpu, memory, video card. Are you really so attached to the case and drives, power supply and monitor? Are you really attached to having a tower? New life, new computer? Maybe time to rethink an Alienware laptop! Traveling, moving around, apartment space, less furniture, you won't have a car... ! Just ideas to a PC guy (two laptops here)