For a three monitor set up you probably can make it work from an integrated GPU. The latest Ryzen 7000 series CPUs now all have iGPUs too, so you are not limited to Intel, for earlier generations, you needed to order specific versions (ending in G) to get a GPU. Similarly for the Intel CPUs be careful, they don't all have integrated graphics, a 12400K will but a 12400F will not for example.
With all that said, the biggest headache you are going to have is making sure that the motherboard will either support three video outputs (I have definitely seen dual many times, but have never looked specifically for triple), or finding one that supports something like daisy-chaining thunderbolt monitors or similar. In my opinion, it's not worth it - you are going to potentially drive up the cost of your motherboard, limit your CPU options, and have your CPU working hard on graphics when you are looking to use those resources for compilation etc.
Instead, I would recommend looking at a dedicated GPU that offers 3x monitor support out of the box. You are going to make your life so much easier by doing so, and I would imagine you can save almost the cost of a budget GPU by being more flexible around motherboard and CPU options.
Of course, GPUs are always a moving target and it will depend on what other purposes you intend to put the GPU to (occasional gaming perhaps?). Right now (March 2023) the best bang for the buck card if you don't care about game compatibility/performance is probably one of the Intel ARC cards. The entry level A380 is available for less than $150 at the time of writing and can support up to 4 displays (3 Displayport, 1 HDMI). You'll need to do a little bit of digging to make sure that works for your particular combo of displays (what refresh rate, resolution do you want to actually run them at?).
