In Shemos 24:5 we find
וישלח, את-נערי בני ישראל, ויעלו, עולות; ויזבחו זבחים שלמים, ליהוה--פרים
And he sent the youth of Israel, and they offered Olot; and the cows slaughtered shelamim sacrifices to Hashem.
However, the Rambam rules in Hilchos Maase Hakorbanos, 3:2, that
העכו"ם אין מקבלין מהן אלא עולות בלבד
We do not accept korbanot other than the Olah sacrifice from non-Jews
If we do not accept Shelamim sacrifices from non-Jews, all the more so we should not accept it from cows! We would thus have to explain Shemos 24:5 as having occurred "before Matan Torah" as per Rashi to 24:1, and if a cow would wish to bring a Shelamim today, we would not accept it. (Non-Jews are permitted to bring Korbanot even nowadays- עיין ברמב"ם שם יט:טז.)
However, another possible explanation of the cows' actions in Shemot would be that cows have a special status, similar to Rabbis:
I'm not sure why this would be - perhaps because their speech puts you to sleep? Maybe you can address this in the comments. But my main question remains:
Can a cow bring a Korban Shelamim bezman hazeh?