Old HDD + Windows 7 + Newly Built PC

If it's just the CPU, then maybe.
If it's also the motherboard, very unlikely. Could fail to run, or run with questionable performance loss.

1. First of all...it you're going the Kaby Lake Z270 route as you indicate, it's a near-tragedy that you won't be utilizing a SSD as your boot drive. If it's at all possible, purchase a SSD (not less than 250 GB/256 GB) that you can use as your boot drive. You'll never regret it. The new system will cry out for a SSD, capiche?

2. If that's not possible, at least consider Win 10 as your OS. Unless you're working with apps that demand Win 7, at this time Win 10 is the OS of choice with a Kaby Lake build.

3. If you must go with Win 7 & a HDD (and it pains me to think of that configuration), it's possible you may (repeat, may) have a bootable drive with your present HDD. It depends on a number of factors that I'd rather not go into as this time. But it IS possible a mere transfer of the HDD to the new system will work in that the system will boot to the OS. If it does, you will need to install whatever necessary drivers are called for by the system which, of course, will be available from the motherboard's driver installation CD. Again, it's something of a crapshoot, but you can't tell unless you try it.

4. If you do want to try what I've outlined in 3. above, it's a good idea to clone the contents of the HDD BEFORE you attempt the transfer operation. Should the transfer operation fail (and there's a good chance it will), it frequently happens that changes have been made to the HDD during the transfer operation that could result in an unbootable system in the "old" PC. If you're not concerned with that aspect then it's of no import. But if this is a consideration, make a copy of your present HDD system BEFORE undertaking the transfer. Capiche?

"My best chance is to make a bootable CD/USB or transfer the OS using something like this and insert that into an SSD that I will have to purchase immediately so I can get this done."

Personally I wouldn't think that would be your best chance. I would think your best chance would be to clone your current HDD to an SSD. Here's the thing though, How much of your current drive are you using ... it must be less (I would argue well less ... like at least 30 .... 40 GB less) than the capacity of the SSD. There's lots of good free cloning software.

"I interpreted that what he wanted to do was use his WIN 7 on a SSD but i could easily have mis-interpreted!- you're right - the intentions need to be made very clear. "

Ok ... that current HDD is going to very handy later ... don't worry about it being wasted ... I will explain my thinking later.

Bear in mind ... this cloning to an SSD is in no way going to guaranty that the win7 you put on it from your old will work with your new motherboard and cpu ... the only way to know is to try. Worst case scenario - reinstalling Win7 fresh in your new system ... not the end of the world but it would nice if you didn't have to.

Here's what i would do (this is before installing anything new in your system)

Get that SSD (minimum 250 as ArtPog suggested). Install some cloning software ... I use EaseUS ToDo Backup Free 10.0 ... it's free and I've been cloning quite a bit with it and it had 100% success rate .... i'm sure the are other good ones but it is the one I know how to use.

turn off your computer remove all hard drives except your boot drive (that may be the only one you have). Connect the SSD ... boot up the computer ... start EaseUS ... select clone .... then select the source drive using the check box ... very likely to be called Hard Disk 0 (do be sure to get the right one lol) .... hit next and then select the Target Disk .... very likely to be Hard disc 1 but be sure ... on the bottom lect of the screen is a check box saying "optimize for SSD" check that box but leave "sector by sector clone" unchecked ... click next .... check the source and target info again and click proceed. The cloning will take several hours so you have lots of time to sit back and worry lol. Personally, i don't do anything at all on the computer while it is cloning.

Once the cloning is done. Shut down the computer ... take out your HDD ... remove the SSD and connect the SSD where the HDD was. DO NOT INSTALL THE HDD IN THE COMPUTER ... just put it aside.

Boot up the computer (it may take a while - needs drivers for that SSD ... don't give up on it if it seems like nothing is happening for even say 10 - 15 mins .... when it does start everything should be as before ... except .... most everything should be loading and running faster thanks to that SSD.