The most natural way in which to interpret this text is to see both the "at him" and "his day" as having the same antecedent, i.e. representing the person the Lord is considering in the text.
However, Hebrew is always highly context-dependent, and it is not infrequent that a Hebrew expression may have a dual meaning, or multiple layers of meaning. According to my Hebrew professors, they believe this is intentional on the part of the Hebrew authors, or the God-given inspiration behind it. This allows the text to be interpreted in a manner most needed or appreciated by a given individual in the particular circumstance he or she may find himself or herself.
That said, it is possible that the text addresses either God's day or the man's day, and while contextually it would be inconsistent for the "he sees" to be the action of the man, instead of God, even this is theoretically possible.
The most natural interpretation, however, would be this:
The Lord shall laugh at him [the man]: for he [the Lord] sees that his
[the man's] day is coming.
It may be worth noting that in Hebrew the masculine third-person pronoun may also serve in the place of the neutral "it"--though in this particular verse, the context clearly addresses a person, and not a thing. However, Hebrew does not distinguish between God and man via pronouns: there is no special pronoun that would set God apart from common man.
Another point of interest relative to the Hebrew text here: the word "day" is singular. This, then, refers to a specific time, because the word "yowm" (H3117) is often used in plural form to indicate a time or an age, such as when addressing "all the days that Adam lived" (Genesis 5:5, KJV). So whether it refers to the Lord's day or to the man's day, in either case it narrows this to a specific day or time.
While ambiguity technically exists, the most natural conclusion, based on the contextual clues of the verse itself, is that the "him" and the "his" of the verse apply to the man, whose "day"--left unspecified as to whether this might refer to his judgment, his death, etc.--is coming.