The Lincolnshire side reached round one for the first time by beating National League Wrexham 3-2 after extra-time in a fourth qualifying round replay.
Stamford, who play in the eighth tier Northern Premier League Division One South, now face League One Hartlepool.
"We're on a bonus for the FA Cup and we're determined to go to America," manager Graham Drury said.
"At the minute we've got nearly £2,000 and if we beat Hartlepool in the next round I'll be saying to the chairman I want another £5,000 to go into that players' pot.
"That's our goal, our dream, to send the boys to America."
Drury, who led Conference North side Corby Town into the first round of the FA Cup five years ago, took over as manager of Stamford last season but was unable to save them from relegation from the Northern Premier League Premier Division.
And he said was delighted to make club history by giving them some cup success.
"When I came into Stamford at the back end of Christmas last year they'd won one home game in 18 months," Drury told BBC Radio Lincolnshire.
"We've turned that round and made Stamford a bit of a fortress. We've lost one game here this season and we've played some top-quality sides.
"We've stuck at it and we've got our rewards, and to take them to the first round proper, I'm a very proud man."

Eastmond, capped six times, previously played for St Helens and the England rugby league team before switching codes to join Bath in 2011.
The 27-year-old made 72 top-flight appearances for Bath, scoring 16 tries, including two last season.
"Kyle has already shown his international class and still has plenty of potential to fulfil," Wasps director of rugby Dai Young said.
Wasps have not disclosed the details of Eastmond's contract at the Ricoh Arena. He had agreed a new deal at Bath in January.
Eastmond, whose last international appearance for England came against South Africa in November 2014, becomes Wasps' 12th signing ahead of the 2016-17 season.
"Kyle is one of the most exciting centres in the Premiership," Young told the club website.
"We're really looking forward to adding his talents to an already impressive group of backs at the club."
Stamford's players are hoping to earn enough money from their FA Cup run to fund an end-of-season trip to the US.
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Wasps have signed England centre Kyle Eastmond from Premiership rivals Bath.