We know that some countries like Israel have lobbies in the US.
Why does USA allow other countries to influence their policy?
What is the reaction of US tax payers to the existance of such lobbies?
Practically all governments in the world try to influence each other. One government investing resources into convincing other governments to be more benevolent towards them is nothing unusual. This is called "diplomacy".
In the special case of the stance of the United States towards Israel, there is also a strong domestic pro-Israel lobby. But there is no reason to believe that this lobby is financed or controlled by the government of Israel. About 7 Million of US citizens are Jewish Americans (depending on how you count). They are on average more politically active than other US citizens. This partisan source claims 85% voted in the last Presidential election, while the overall turnout was just 55.5%.
Not everyone who is Jewish is automatically pro Israel, of course. But organizations which defend the rights of Jewish US citizens, like the Anti-Defamation League and the American Jewish Committee, have repeatedly issued statements encouraging solidarity between the United States and Israel.
Regarding how the US public in general thinks about Israel: While Israel is a controversial topic (the anti-Zionists in the United States are also rather vocal), the majority of US citizens are pro-Israel. When Pew Research asked US citizens about their opinion regarding the Israel/Palestine conflict in 2014, they found that:
51 percent of Americans said that they sympathize more with Israel in the “dispute between Israel and the Palestinians.” Only 14% said their sympathies are more with the Palestinians. Another 15% said they sympathize with neither side, and 3% with both sides.