Absolutely!

The Ethernet port on your eMachines is part of the motherboard.
It is attached directly to the motherboard.

The eMachines T3626 Desktop PC, is supposed to use an
ECS MCP61-P (AM2) motherboard. (Cannot verify)
(Elitegroup Computer Systems. (AM2) refers to the processor socket )

http://www.ebay.com/itm/MotherBoard-eMachines-T3626-T5062-T5234-W3622-/320528791641

(Also scroll down for a larger photo of the motherboard )

The Ethernet port is in the group signified by the letter A, on the left.
The group signified by the letter A, is in the I/O area.

Input/Output area. This is where Input, and Output devices are attached, TO the motherboard. (Using the motherboard's chipset)

Now look down the motherboard, to the two long white slots.
These are PCI slots. Signified by the letter I ( i )
If one of these slots is not being used, plug in an Ethernet Card, that is PCI.

If both of these PCI slots are being used, use the small white slot above them, signified by the letter H.

This is a PCI-Express x1 slot. If it is available, use it.
Use an Ethernet Card designed to fit, in a PCI-Express x1 slot.

Example;

PCI Ethernet Adapter Card,

http://www.tigerdirect.com/applications/SearchTools/item-details.asp?EdpNo=771339&CatId=1175

PCI-Express x1 Ethernet Adapter Card,

http://www.newegg.com/Product/Product.aspx?Item=N82E16833106033

[ Note*
The PCI-Express x16 slot that you have, (Orange in color, above the white PCI-Express x1 slot), is for a graphics card.

It is ONLY for a graphics card.
The PCI-Express x1 slot is NOT used for a graphics card ]

For additional questions please post in a Comment.
(If you would like the proper procedure for installing a Ethernet Adapter Card, in your eMachines T3626, also state in a Comment)

Regards,
joecoolvette