Trump is, as he often is, speaking in a manner that conveys no clear meaning. Taking his phrasing literally, he is discussing Exxon sending $25m to him personally, for the purpose of him spending it in his campaign. But it can also be taken as an imprecisely worded description of Exxon sending $25m to his campaign. A politician's campaign and the politician themselves are separate entities, and there are very different rules about money going to them. However, any money given to a person for the intent of it going to a candidate's campaign is required to follow at least as many conditions as money given directly to a campaign. Since corporations are prohibited from giving money to candidates' campaigns, either interpretations describe a wildly illegal situation:
Campaigns are prohibited from accepting contributions from certain types of organizations and individuals. These prohibited sources are:
Corporations, including nonprofit corporations (although funds from a corporate separate segregated fund are permissible)
But corporations are often described as giving to a candidate's campaign when in fact they are engaging in independent expenditures, so Trump may be following this habit. A corporation is allowed to spend money advocating a particular candidate, as long as they do not coordinate with that candidate. If Trump asks a corporation to spend money, that counts as coordination, so again, wildly illegal.
Coordinated means made in cooperation, consultation or concert with, or at the request or suggestion of, a candidate, a candidate’s authorized committee, or their agents, or a political party committee or its agents.
nick012000 brings up in the comments the fact that sometimes when a corporation is said to be contributing to a campaign, it is actually employees that are doing so. I think it would be a stretch to interpret this hypothetical as referring to that, but then with Trump, who knows. He certainly couldn't be (legally) asking for the particular Exxon executive he is speaking to to contribute $25m.
During the current two-year election cycle the limit for contributions by individuals to federal candidates for President, the U.S. Senate and the U.S. House of Representatives has increased to $2,800 per election.
Thus, it would take nearly ten thousand Exxon employees to legally contribute $25m to Trump. There's complications such as soft money versus hard money, etc., that could reduce the number of people needed to get to $25m, if not all the money is going directly to Trump's campaign, but there's no way a single person can legally give $25m. Getting 10,000 people to contribute $2800 each is technically legal, but could easily get one in hot water if one is found to have improperly induced them to do so, and it's difficult to imagine getting that level of participation without some sort of incentive.
There is also the fact that this would be bribery, and that charge would remain even if Trump were to dispense with the whole campaign thing, and just ask Exxon to send $25m to his personal bank account.
(2)being a public official or person selected to be a public official, directly or indirectly, corruptly demands, seeks, receives, accepts, or agrees to receive or accept anything of value personally or for any other person or entity, in return for:
(A)being influenced in the performance of any official act;
shall be fined under this title or not more than three times the monetary equivalent of the thing of value, whichever is greater, or imprisoned for not more than fifteen years, or both, and may be disqualified from holding any office of honor, trust, or profit under the United States.
As a side note tangential to the central question of election law, Trump has limited ability to hand out permits. If Exxon faces state or local permit requirements, the federal government can't help with that (although it can apply pressure). If Exxon wants to drill on federal land or offshore, then there may be federal permits required. Those would not be handed out by Trump personally, but by officials of the appropriate department. It would be highly irregular for the president to order an agency to issue a permit, but with this president, that would not be out of the question.