Is powering a sound system from a computer's power supply a bad idea?
A modern ATX12V supply will easily meet your power spec.¹

Still, there are a number of problems with PC power supplies that make them less than ideal for audio:


Switching power: PC power supplies have always been switch-mode types, rather than linear. Originally, this was simply because of the amount of power that a computer required.²

There ain't no such thing as a free lunch, so the cost of the higher efficiency of a switcher is that it is electrically noisier than a linear. This isn't a big problem for digital circuitry, especially back in the days of 5&nbsp;V logic, where a few tens of millivolts of noise was easily ignored by the gate thresholds.

For audio, though, noise is a problem, since it can indirectly affect the produced waveform.

I say "indirectly" because almost all amplifiers have some amount of PSRR, which rejects rail noise.³ PSRR isn't perfect &mdash; there is no such thing as a circuit with infinite PSRR &mdash; and it tends to drop as frequency increases, often starting to drop in the audio frequency range.

When deciding if you have too much rail noise or too little PSRR, you must consider the sensitivity of the human ear, which has an amazing dynamic range.

Example: Your spec says 12&nbsp;V &plusmn;5%, so that allows up to 600&nbsp;mV of noise and ripple (N+R) on the rails, assuming no DC inaccuracy. Let's say it's really only averaging 100&nbsp;mV of N+R. Let's also say that at the frequency of the noise on your PC power rail, PSRR is 60&nbsp;dB. Let's further say that your audio system needs 3&nbsp;Vac to the speaker to reach 90&nbsp;dB&nbsp;SPL.⁴ 100&nbsp;mV is about 30&nbsp;dB down from 3&nbsp;V, so adding the 60&nbsp;dB PSRR, you find that the noise should be right at the threshold of inaudibility: 90&nbsp;dB&nbsp;SPL operating maximum - 30&nbsp;dB noise level - 60&nbsp;dB PSRR = 0&nbsp;dB&nbsp;SPL noise output, the limit of human hearing.⁵

Barely inaudible, mind. I contrived that example to give that result. Maybe your amplifier has lower PSRR, or higher inherent higher noise, or higher coupled noise from the PC power supply, or a higher output voltage requirement. The point is, PC power rail noise could easily become audible. If this system is used in a quiet room or with closed headphones, so that there is nothing to mask the noise, your choice to use a PC power supply might effectively ruin the project.

Gain and amplifier stage design also play into this. Some amplifiers do their work in multiple stages, each affected by the power supply. If you get -90&nbsp;dB noise in an early stage which is followed by a gain stage, the noise gets amplified, too. When the amplifier's volume is turned all the way up, it could be 10&nbsp;dB&nbsp;SPL or more.
Shared rail: Many other parts of the PC will be running from that powerful +12&nbsp;V rail. Those components will be injecting their own noises into that rail, adding to what the power supply itself produces.

Some PC power supplies have multiple independent 12&nbsp;V rails, which would mitigate against this. You may be able to find an unused rail, typically intended for powering high-end graphics cards.
Ground noise: In a sense, this is just an extension of the above, but it has special consequences when it comes to audio systems.

The ground in an electrical system never has zero impedance, so noises shunted through components to ground don't go to zero. All electrical components produce noise, Johnson noise if nothing else.

Another way ground noise occurs is called "ground bounce," when a component's dynamic current draw creates current-modulated IR drops in the ground plane. 

Still another way the ground gets noisy is because components don't have infinite PSRR, so it couples some portion of the power rail's noise to ground.

Now, what happens to all that ground rail noise when it gets to the audio system? Your speakers are likely to be referenced directly to this PC power ground, for one thing. That means all the PSRR in the amplifier does diddly-squat for reducing this noise. Ground bounce bounces the speaker, which turns it directly into sound energy.

Is this fatal? Depends on how much ground rail noise there is.
Symmetric power: Some audio circuits require not a single-ended power supply as in your 12&nbsp;V spec, but instead &plusmn;12&nbsp;V or similar. PC power supplies do still include a -12&nbsp;V rail, but because almost nothing makes use of it these days, it's always one of the weakest rails in the system, if not the weakest.

Yet, if you don't make use of it, you have to use several hacks to get audio to work on a single-ended system, which themselves harm the sound quality. Capacitive-coupling of the audio signal, for instance, adds distortions from capacitor imperfections to the signal.⁶
Shared RFI environment: PC-based power implies that the audio components will be living inside the RF noise bath inside the computer chassis. That means they're subject to having that RF noise coupled into the audio circuitry, which can result in audio flaws.




Footnotes:


If you're talking instead about scavenging old power supplies from dead computers, you might run into trouble, as the +12V rail was often one of the least powerful, rather than the most powerful.
In the early days of PC computing, the CPU unit typically required at least 100&nbsp;W of continuous power, which would require a pretty big and hot linear power supply. Today, you can get quite capable low-power PCs that run under 10&nbsp;W, which could reasonably be run from a smallish linear power supply. You still won't find that done commercially, though, since there are now various regulatory and market pressures that require efficient power supplies.
There are a few uncommon types of audio amplifier where rail noise directly wiggles the output, so for this sort, you absolutely require a stable, quiet linear supply.
90&nbsp;dB&nbsp;SPL is a pretty good value for "loud enough" in an audio system, unless you're building concert hall or stadium sound systems.
By definition, 0&nbsp;dB&nbsp;SPL is the human threshold of inaudibility.
Notice that that article doesn't even try to cover electrolytics, which are terribly nonlinear. Yet, a lot of audio systems do use them for audio coupling.