Possibly, although probably not immediately. As Stančikas points out, the USA is a signatory to the Treaty on the Non-Proliferation of Nuclear Weapons. Article I of this states,
Each nuclear-weapon State Party to the Treaty undertakes not to transfer to any recipient
whatsoever nuclear weapons or other nuclear explosive devices or control over such weapons or
explosive devices directly, or indirectly
which seems to rule out the possibility of the US president authorising the transfer of nuclear weapons to Ukraine. However, there are two important caveats:
For all of the time that the USA has been party to the treaty, it has also been sharing nuclear weapons with other NATO member states (who are also treaty signatories), currently: Germany, Italy, Belgium, the Netherlands and Turkey. The justification for this is that the weapons haven't really been transferred to those countries, because they are under the control of US forces, but it has raised criticism from other NPT signatories. Nevertheless, this nuclear sharing framework gives some indication of the how the US president could circumvent the NPT if they wished.
Article X of the treaty provides a mechanism for states to exit the treaty.
Each Party shall in exercising its national sovereignty have the right to withdraw from the
Treaty if it decides that extraordinary events, related to the subject matter of this Treaty, have
jeopardized the supreme interests of its country. It shall give notice of such withdrawal to all
other Parties to the Treaty and to the United Nations Security Council three months in advance.
Although the USA ratified the treaty with the consent of congress, it is possible that the US president could withdraw from the treaty without congressional approval. The case of president Carter and the Sino-American Mutual Defence Treaty is an example of this happening in the past. In that instance, some senators opposed Carter's decision and brought the case to the Supreme Court (Goldwater v. Carter) but the court declined to rule on the matter, on the basis that it had no jurisdiction. It is possible that the court would do the same again, if a US president decided to withdraw from the NPT, meaning that three months later the president would probably have the authority to supply Ukraine with nuclear weapons.