Okay, minor correction on flashbulb types:
M bulbs are those with a "midget" base, round with a groove at the end and a central contact. There were M2 and M3, and there was once an M5 as well (same light as a #5, but with a midget base), though I haven't seen them still around the way M2 and M3 are. The #5 and P25 are bayonet base, similar to a car taillight bulb. There were also AG-1 and AG-3 "baseless" bulbs ("AG" for "all glass"), with the latching form molded into the glass envelope and the contact wires just wrapped around the glass at the bottom. M3, #5, and P25 are Type M bulbs, 20 ms ignition time and 25 ms burn time, for use with M synch. M2 are classed as "MF" bulbs along with AG-1 and AG-3; a faster igniting, faster burning bulb (IIRC, 15 ms ignition and 20 ms burn), type MF bulbs can be used with X sync at 1/30, while Type M require 1/15 to get the full burn on X sync. Most M synch shutters made after about 1955 are actually MF, and work perfectly with M2 bulbs as well as M3 (they can cut the tail off the M3 burn if the speed is set too fast, but you normally shoot at 1/30 anyway with a bulb). Type FP bulbs have ignition of 30 ms, and burn of 125 ms, so as to provide even light during the entire travel of a focal plane shutter (of the Speed Graphic type); the most common and the only ones I've seen on eBay recently are #6, same guide number as a #5 but with the long burn. There was also a Type F bulb -- I recently saw a box of them go by on eBay, looking a bit odd because they're filled only with gas, no magnesium wire -- they had effectively instant ignition and something like 10 ms burn, and were for use with F synch (though they also work with X synch and shutter slower than 1/100), found on only a few press camera shutters (the bulbs were mainly of use to stop action, and vanished about a week after xenon flashes came out because only press photographers ever used them, and they switched to strobes as fast as they could get their shutters converted to X).
Now, you may run into trouble attaching that Kodalite IV -- I don't recall if it has the screw and post terminal the Hawkeye Flash accepts. If it does, modern alkaline cells that fit the clips will work fine; the old "photoflash" carbon-zinc cells had a high pulse current at the expense of overall capacity, but modern alkalines have still higher pulse current (meaning they'll fire the bulb just as positively) but also much higher total capacity -- they're just plain better. If the Kodalite IV doesn't mate the Hawkeye's flash terminals, keep an eye on eBay, I see the Kodak flash units with screw and pin terminals every now and then, going for $5. With no electronics in them (most didn't even have a capacitor, depending on big, low-voltage cells to fire the bulb instead of a small battery at higher voltage with a capacitor to provide higher current for ignition), even if they're badly corroded you can cut out the really bad bits and solder in brass or copper strips to replace the parts you removed.
If you can get them to answer e-mail, Cress Photo (www.flashbulbs.com) has much better prices on bulbs than J&C Photo, and you can easily beat both by setting up a search on eBay and being patient. Do watch the shipping costs on eBay, though...