Calculate the minimal MOSFET source current needed for 12 TTL MOSFETS
You were probably looking at this chart when you got that &quot;\$V_{DS}\$ to \$I_{D}\$, depending on \$V_{GS}\$&quot; business that gave you 9 amperes:

You missed the note in the corner that says &quot;20 microsecond pulse width.&quot;
What that chart says is that if you provide a 20 microseconds pulse to the following circuit then there will be 20 microsecond long pulse of the current given in the chart.



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That's not a very useful circuit, and that high current rating isn't that useful.
For continuous current, you need to look at the &quot;Absolute Maximum Ratings&quot; which says 2.7A at 25 C.  You also need to look at this chart which gives you information on what it is safe to do for various current and voltage combinations:

What those charts are trying to tell you is how much current may be passed through the transistor under various conditions.
All of that still doesn't solve your real problem, though.
The real problem is that you are doing this with your transistors:



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That shows a mechanical relay rather than the SSR you intend to use, but it works the same.
There are two problems with your circuit:

It takes two connections to the relay to make it switch, and you've only provided one.
You have your transistor set up to short circuit the power supply.

Your relay circuit needs to look like this:



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The relay (whether SSR or mechanical) will limit the current through the transistor - no more than a few 10s of milliamperes will flow for the SSR, maybe a couple of hundred milliamperes if it were a mechanical relay.
Here at last is the answer to your question:

How much current should your 12V power supply be able to deliver, assuming that all relays may be activated at once.

If you are driving 12 relays, then the answer is &quot;12 times the control current of one relay.&quot;
Your linked SSR datasheets both say &quot;less than 10 milliamperes of control current.&quot;
A 12V power supply capable of supplying 120 milliamperes would be enough to drive all 12 SSRs at once.

You mention in comments that you may need to drive loads directly with the IRLL014.
In that case, your 12V power supply would need to supply enough current for the loads.  Given that the IRLL014 can only handle a couple of amperes continuously, your power supply would need to be able to supply something over 24 amperes continuously.
The circuit would also be a little different:



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