Rich people aren't spending all their money on stuff, they're investing it. If wealthy people were to take all that wealth and start buying stuff (themselves or via redistribution through taxes to UBI), there isn't magically more stuff, so the price of stuff goes way up => inflation.
Investing is just lending money to people who need to buy something, such as capital. That money is still spent. In fact it needs to be spent on production, in order to produce returns.
You can’t just store wealth in a vault and have it grow.
Most companies can easily ramp up production to meet increased demand. It's not magic, it's the modern world.
Most general goods companies are demand constrained, that's why they need to do marketing campaigns to make more sales. It's only if they had too many orders and couldn't grow production that we'd have inflation. So there may be some inflation immediately but not long term.
