In order for it to cause inflation, there has to be a net increase in dollars put into the economy. If they're just putting the dollars into household accounts rather than sending it down through .gov and NGO paychecks, there's no difference than status quo.
In the simplest of examples: Every dollar given out has to be printed. The govt is running a $2 trillion budget deficit right now. That means there is not enough money to pay for what we have budgeted and plan to spend. So in order to fund the budget they have to print additional money and add it to the debt. The savings DOGE is finding are unspent money that they were going to have to print and put into circulation to pay salaries, buy materials, and fulfill, contracts. Every dollar doge is finding are unspent funds, the money for which is not sitting in the treasury. In order to pass it out to taxpayers, they will still have to print it. Every time they spend money in excess of what they take in, they are taking it on to the debt very new dollar printed and dumped into the economy, increases inflation.