It seems really weird to be unable to buy clothes with cash. Maybe they are actually in a store that is too expensive for them?
It seems likely here that the clothes Komi was picking out exceeded the intended budget by a bit but he was just excited to use the outfits his daughter picked out for him and treat her to what she wanted. It's also possible that Komi-dad wasn't carrying much cash to begin with because it's not exactly wrong to use a credit card on normal purchases you can definitely afford to pay off before even incurring interest.
Consider the following. Cash: can take up more space the more you have, if lost or stolen... that's it, it's gone, sucks to be you. Credit card: same amount of space at all times, if lost or stolen you can just cancel it as soon as you notice it's gone, potentially minimum damage to your funds.
Also, if Japan's working things out similarly to how we do with the whole credit thing, using a credit card responsibly on things you know you can pay off immediately when the bill rolls in on you is an excellent way to improve your credit score type dealy.