Most countries try to influence each other.
That can be in the form of public and private diplomacy of the embassy and consulates, talking to parliaments, governments, and the public. It can be in the form of public diplomacy from the home country, writing press releases designed to affect a target population in a certain way. (Compare the Ukrainian president, quoting Churchill to the Brits, talking about walls to the Germans, etc.) It can be by hiring public relations companies at home or abroad.
That is generally considered legitimate, at least if public statements disclose their origin.
Some countries try to pressure each other.
That depends on what levers those countries have. They might raise tariffs, or promise to lower them. They might give favorable or unfavorable prices for exports like oil, or refuse to sell at all. (Compare how Russia deals with their gas contracts right now.) They might provide financial guarantees to encourage trade.
That is sometimes considered legitimate. Breaking contracts is not OK, offering favorable contracts to some might fall under trade agreements a country might have signed, but generally offering good deals to friendly countries is permitted.
As to the specific case:
- Saudi Arabia has oil, and it can influence other countries with oil.
The US has their own sources, but they are still affected by changes in global market prices and supplies. That alone makes Saudi Arabia more influential than, say, Namibia.
- Saudi Arabia has money.
Having money to spend and invest is always a better position than asking for handouts.
- Saudi Arabia is opposed to Iran.
For historical reasons, the US is opposed to Iran. So is the American ally, Israel. Saudi Arabia is one of the Arab countries that may put their hostility towards Iran over their hostility towards Israel.
- Relations become a habit.
Even if objectively speaking, Saudi Arabia is not an obvious ally for the US, it has been an ally for a long time. Changing that would mean overturning established policies in both countries.
That being said, I believe that your question overstates the situation. I believe that the US government does find it convenient to work with and through Saudi Arabia, which shields Saudi Arabia from some but not all backlash their actions would otherwise bring. They are not as influential in Washington as the Canadians, or the UK, for instance.