displaygpumacbook pro
I have 4 Dell 1708FPT monitors (https://www.amazon.com/gp/product/B0065638JS) which use either DVI or VGA to connect.
I am wondering what adapters I would need to get all 4 monitors powered by my MacBook Pro 15 inch early 2013 laptop running macOS Sierra?
There are many types of DVI adapter, I could type a load of stuff or just point you at Wikipedia.
The key point is that there is no such thing as a DVI adapter. It's just a generic term that covers the overall outline of the cable end, but what you really have is either a DVI-I or DVI-D adapter, in either Single or Dual link capability. Or it may be a DVI-A. Or there are others...!
It's possible that your splitter cable must be of the dual link types, characterised by the longer horizontal bar pin, and your Thunderbolt adapter may or may not be. Shouldn't matter which type, either I (Integrated with analogue signal in the pins above/below the horizontal bar in addition to digital) or D (Digital only) as I imagine you are not subsequently adapting them down to DSUB further down the line, the only thing that matters is if they are both dual link.
So it looks like you need a Thunderbolt/MiniDisplayPort to DVI Dual link - the official apple one is the digital only one without the analogue pins (DVI-D Dual Link), and thus should work, if you are worrying that you do not have the same pinout (i.e. you have DVI-I Dual Link on your splitter, and it looks as though the analogue ones have nothing to connect to) then don't worry, it won't need them.
EDIT - Additional Info
After the comments, I have decided that you have a DMS-59 connector that looks like this:
So what you really need is a MiniDisplay > DMS-59 adapter, and my google-fu suggests that this is not looking like a possibility, I think you would be better served abandoning this cable, and looking for a total replacement, Mini Display Port > 2 x DVI (Single is fine, I or D if you have 2 ports). This may also be a pain to find, and you may end up needing a MDP/Thunderbolt to Dual Link DVI, followed by a Dual Link DVI > 2 X Single Link DVI splitter cable!
The mini DisplayPort to DVI adapter will work great, so your 27" monitor is in.
The VGA monitor is problematic. HDMI and DVI share almost identical digital video signals, so any HDMI to DVI adapter (like the included one) has few (if any) conversion chips in it and the output DVI port will be DVI-D (digital only). DVI to VGA adapters also don't have conversion chips because they use DVI-I or DVI-A ports (A is analog only and I id both analog and digital): they take the DVI analog signal and move it straight to the VGA port. Unfortunately, that means that you cannot go from HDMI to VGA via DVI. You'll need a more expensive converter like this or this.
tl;dr: You can't go from HDMI to VGA without a converter; see the links for examples. For the dual-link DVI you need, the adapter you mentioned that will use the Thunderbolt port will work excellently.
Best Answer
Hmmm, this is an interesting question because the monitors you want to connect to are only capable of a maximum resolution of 1280 x 1024.
Apple's official stance
Officially these MBPs can support up to one HDMI display (at 1920x1080 pixels) while simultaneously connected to a Thunderbolt display (at 2560x1600 pixels), OR being connected to two Thunderbolt displays (both at 2560x1600 pixels).
Unofficial real-life scenarios
However, unofficially you can exceed this. I'm not aware of anyone who has tried connecting these MBPs to four displays at a lower resolution. What I have seen instead are the following scenarios with the mid-2012 Retina MacBook Pro which uses the exact same setup (NVIDIA GeForce GT 650M and Intel HD Graphics 4000):
I've also seen users connect three external displays while in clamshell mode. However, this resulted in the MBP's fans spinning at more than double their usual speed when watching videos etc. Not something I would recommend.
Your unique scenario
It's not clear from your question whether you were wanting to use the MBP in clamshell mode. If I assume you do, then you're in a situation where you want to use four external displays with a total of 5,242,880 pixels which is less than the official combinations supported by Apple. So the only problem you're likely to face is how you'd physically do this?
I see the following solutions for connecting four of these particular displays:
Personally I would not recommend Option 4 or Option 5. Not only do these options use up your USB ports, they also double up on using an adapter that will have limitations. By limitations I mean the bandwidth provided by your USB ports will not work well with many games or anything that is graphics intensive. However, if you're just using MS Office or doing some basic web browsing, you should be fine.
Below are some links that may be useful:
Finally, you may find the Matrox TripleHead2Go DP worth considering.