I'm so sorry to hear about Logan Coffey. TT makes excellent, although pricey, equipment.
Now, before I say anything about modular vests, let me state the obvious: I don't do SAR work and I don't know what most SAR people carry. However, I have some familiarity with military equipment, both modern and historical, so maybe I can be of a little bit of help.
As far as MOLLE (Modular Lightweight Load-Carrying Equipment) goes, it's an upgrade to the old ALICE (All-purpose Leightweight Individual Carrying Equipment) system.
ALICE gear attaches using ALICE clips, which are metal clips that slide open and closed to attach pouches and equipment to belts and other places that have webbing for this purpose.
MOLLE gear uses webbing straps instead of metal clips. The straps are looped through the webbing and secure at the bottom with a lift-a-dot type fastener.
Both of these styles of attachment have pros and cons. ALICE clips tend to start wearing out with use and sometimes come undone easily. In rough terrain, that can lead to pouches coming loose or getting lost. MOLLE gear experiences similar issues, especially if snaps are not secured properly, and can also come off. For additional safety, the military tends to secure pouches with 550 cord, 100mph tape, or (in the realm of making a simple thing complicated), MALICE Clips which are made by Tactical Tailor.
Most chest rigs like the one pictures in the original post and the ones used by the military are set up for personal preference. That's made pretty easy today since you can buy the rig and then add various MOLLE (or ALICE) pouches to suit your tastes and purposes.
A basic rig for carrying can be had in many different styles, colors, and widths. Most companies carry them in solid black, solid coyote brown, solid OD green, and various camo patterns. There are also companies that make custom chest-rigs with sewn-on pockets, usually based on your descriptions and measurements, and there are websites providing materials and instructions for do-it-yourself chest rig building.
Below are a couple of links to sites that sell MOLLE gear, ALICE gear, as well as replacement / repair materials:
DIY Tactical - http://www.diytactical.com
(go to "store")
Cheaper Than Dirt - http://www.cheaperthandirt.com
Sportsman's Guide - http://www.sportsmansguide.com
Blackhawk Tactical - http://www.blackhawk.com
Brigade Quartermasters - http://www.actiongear.com
Diamondback Tactical - http://www.diamondbacktactical.com
Galls - http://www.galls.com/military/
Hope the links are useful!