Art II, Sec 2, Cl 1 of The Constitution says of the president "and he shall have Power to grant Reprieves and Pardons for Offenses against the United States, except in Cases of Impeachment". The Constitution does not state any further restrictions on the presidential power, and there are no statutory limits, because, as observed in Ex parte Garland, 71 U.S. 333
The power of pardon conferred by the Constitution upon the President
is unlimited except in cases of impeachment. It extends to every
offence known to the law, and may be exercised at any time after its
commission, either before legal proceedings are taken or during their
pendency, or after conviction and judgment. The power is not subject
to legislative control.
Accordingly, presidents have written pardons on and with numerous different writing instruments. Insofar as there are no constitutional restrictions beyond the aforementioned, it is not required that a pardon be signed, or that it be on a tangible semi-permanent medium. It is impossible to know what such a pardon tweet would look like, beyond the limit on length.
The one place where a presidential signature is required is under the Presentment Clause, that when presented with a bill that passed the two houses of Congress, "If he approve he shall sign it". Such a signature need not be actually written by the president, it may and often is written by an autopen. The question of the legality of autopen signatures for bills has not been presented to SCOTUS, but the Department of Justice has issued an opinion (July 7, 2005) that it is legal.
I have failed to locate a repository of presidential pardons (the actual documents), so I do not know if, so far, all presidential pardons were written down and signed, though I would expect it to be so. The Arpaio pardon was signed (or auto-signed), likewise Obama's final mass-pardon on Jan 17, and previous Obama pardons were, but pre-Obama, DOJ does not provide any document.