Is it grammatical to say:
So happy I am today!
I am so happy today!
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*So happy I am today!
So happy am I today!
In the Original Poster's example, (1), the complement of the verb BE has been preposed. The normal phrase order would be:
It is perfectly grammatical to prepose the complement of the verb BE in this way. However, when we prepose phrases starting with the degree adverb so, this triggers obligatory subject-auxiliary inversion. In other words the subject of the sentence and the auxiliary verb change places:
In order for the Original Poster's sentence to be grammatical we need to change the order of the subject and auxiliary verb BE, as in (2):
This would normally be considered a type of exclamation. The change of phrase order gives it a certain literary style. You would be very unlikely to hear it in normal conversation!
Yes, "So happy I am today" is grammatical. The word order is unconventional. The ordinary, prosaic word order would be "I am so happy today." The customary poetic inversion would have "am I": "So happy today am I", "So happy am I today", etc.
Because "So happy I am today" has unconventional word order, it creates unusual and strong emphasis. To a listener, the unexpected order is jarring at first. This forces the listener's mind to find a way make grammatical sense of the sentence, and that suggests that you must be emphasizing something different than the usual word order would emphasize. Usually unconventional word order puts the emphasis on the last word in the sentence, but in this sentence, I think the emphasis goes on "so happy". It appears that you are emphasizing your particular degree of happiness as unusual or remarkable.
The unusual word order might lead a listener to parse the sentence differently than you expect. The word "so" can mean "consequently" or "very" (and a few other things, too). At the beginning of a sentence, it usually means "consequently". So, a listener might understand your sentence to mean the same as "Consequently I am happy today," perhaps referring to a cause that you explained earlier, or perhaps leaving the cause mysterious. Of course, with the unconventional word order, that would be "Consequently, happy I am today."
When you say the unconventional sentence out loud, the word "am" needs to be drawn out longer than usual—longer than in "I am so happy today." In "So happy I am today", the word "am" should last as long as the syllable "-day". Without that, a listener may think that you misspoke, even though the sentence is grammatical. You can prevent the parsing described in the previous paragraph by drawing out "So" for the same amount of time as "am" and "-day".
In musical notes, the rhythm of "So happy I am today" would probably go something like this (not exactly):
♩ ❘ ♪ ♪ ♪ ❘♩ ♪ ❘ ♩
When you use unusual word order, rhythm becomes more important for maintaining comprehensibility.
As you've seen in the other answers and comments, some people argue that "So happy I am today" is ungrammatical, on the basis of explicitly defined rules about exactly when and how inversion must happen in English. John Lawler said that to be grammatical, "it has to occur frequently in the speech of natives, and it has to be rule-governed." I think that's wrong on both counts. Poetic speech gets much of its expressiveness from the fact that it uses forms that do not occur frequently in speech, and I understand English grammar to work something like English common law: not by rules, but by precedents—by echoing and varying previous speech, bending and adjusting it to the unique concerns and pressures of each case, in ways that no finite set of rules could ever fully capture.
Here's an illustration of how contextual pressures can make your sentence more correct than the customary inversion:
So happy I am today!
And so happy I'll be tomorrow!
Are you watching my feet? So happ'ly they beat!
So lightly they step, these engines of pep!
I've permanently stamped out sorrow!
This verse draws upon other elements of the language, such as the fact that the contraction "I'll be" doesn't ordinarily get inverted, as well as rhythm and rhyme (and repetition), to make the non-inverted constructions sound normal. For example, "So lightly do they step" would correspond to "So happy am I", but "So lightly do they step" would be a grammatical error (or at least very awkward) in the sentence above, so it further reinforces "So happy I am".
But the verse illustrates an important grain of truth in the other side: "So happy I am today" does not occur in ordinary speech. It's poetic speech. It can easily sound ungrammatical if you don't support it with other cues, like prosody, juxtaposition with a parallel grammatical form (like "I'll be" above), and—especially—a situation or meaning that calls for unusual speech. Figuring out what those other cues need to be in order to establish grammaticality requires a lot of intuition, which can only be learned from long experience. And of course, different listeners have different levels of willingness to stretch conventional forms.