My dad used the expression 'hoo-wee!' a lot when I was a kid. (That is what it sounds like.) For example, if we were using the grill and it flared up he would say "hoo-wee!", and I love saying it but I'm not sure how to spell it.
Since the expression is an interjection—a cry of surprise or delight, really—many dictionaries (including Merriam-Webster's, evidently) don't consider it a standard word; in any event they don't list it. As Mari-Lou A points out, several online dictionaries—Wiktionary, YourDictionary, and Urban Dictionary—have entries for hoowee. But the Urban Dictionary also has entries for whoo-wee, whowee, and oowee—and a Google Books search turns up numerous examples in which one or another of these spellings of the term is used to express essentially the same thing.
From Ken Kesey, One Flew Over the Cuckoo's Nest (1962):
The motor chugged and died, chugged again like it was clearing its throat, then roared full on.
“Hoowee! There she goes. Pour the coal to 'er, George, and all hands stand by to repel boarders!"
From Jerry Tidwell, One Family's Journey Through Time (2009):
John W. always punctuated his sentences with Whoowee when he was excited or wanted to make a point. Aunt Mag, the youngest child, was sitting in the back of the wagon and said “Whooowee, I can ride anywhere you can drive!”
From Daniel B. Frank, Deep Blue Funk and Other Stories (1983):
"Y'all ain't seen nothin'," Karl exclaims, "till you've seen my grandmother, boy, whowee! Could she raise the dead with a whuppin'! I remember this one night, my grandmother wanted me to do something and I said, 'no.' Well, I did 'n I saw her start to come after me, so I grabbed the belt off the wall so she couldn't use it on me. And do you know what she did? She went upstairs 'n got an old pipe 'n come after me with it. Whowee, did I run. Run right out the house into the snow with no shoes on!"
And from Clarence Major, Dirty Bird Blues (1997):
"Oowee! Lord, have mercy, look who's here. Speak of the devil," said Trudy, grinning from ear to ear, glancing up at him, as she continued working at an elderly woman's short gray hair.
In the absence of any firmly established authority on the subject, you can decide for yourself which spelling of this expostulation you want to use—and so can everyone else.
Following up on Joe Blow's suggestion in a comment above, I also found Google Books matches for hoohee, hoo-whee (and hoowhee), whohee, whoohee, whoowhee, whowhee, woohee, and woowee. From James Post, Summer Chickens (and a Velvet Web) (2002):
"Go for it, Bro!" he exhorted the doubtful-looking Barham. “Hoohee, that is some bad blow! Go on, get your beak in the bowl. It ain't like we're poor people, you know what I mean? Get down!”
From Carla Stewart, Stardust: A Novel (2012):
Behind me, Bobby Carl breathed on my neck. “Hoo-whee! That girl sings almost as purty as you. What're you gonna sing when your turn comes?”
From Michelle Sodaro, Whatever You Make of It (2011):
"Sure, Jac, my boy. I sure hope I remember what we were talking about — whohee sometimes my memory short-circuits on me.” Henry was smiling as usual.
From Bernard Veale, The Wide Clear Sky (2014):
“Whoohee! He got Yellow Jack? That knocks all tarnation outta a man. He gonna be lucky iffen he ever gets to California.” Billy commented.
From Marie Bostwick, A Single Thread (2012):
"Anyway, Heather calls and says that they want to put us on television! Our own show! Quintessential Quilting with Mary Dell and Howard! Can you believe it? Whoowhee!”
From Sarah Shankman, Keeping Secrets (1990):
"Whowhee! That's some brass. They'd have a fit, right?"
From Tom Hron, Whispers of the River (1996):
"Woohee ... I guess I got to raise you two thousand, how you like them apples?
"Woohee—mighty big bets to call. You fellows rightly put your chits in . . . and it's only fair that I go all in to call."
And from Mary Monroe, The Butterfly's Daughter (2012):
When they crossed the Red River into Texas, she honked the horn and let loose with a piercing whistle. “That's the Red River, baby!” she shouted, leaning on the horn. “Woowee! That's home!”
No wonder the people at Merriam-Webster are staying far, far away from this issue of orthography.
Write it any way you want! There is no standard spelling of this interjection as illustrated by Sven Yargs. Thankfully, this one place were English still allows improvisation. Yeehaw, go for it! (this would have been a comment to the above answer but I don't have enough reputation!!!)
Onomatopoeic (or Echomimetic) words like this are words that imitate sounds. You'll find that in some cases, the these words converge to more frequently used spellings, but there is no hard rule that you must follow.
In 100 Mostly Small But Expressive Interjections, Mark Nichol (Daily Writing Tips) lists many, along with their spelling variations, illustrating the point that there is seldom a consensus for this sort of thing.
Woo and woo-hoo (and variations like yahoo, yee-haw, and yippee) indicate excitement. (Woot, also spelled w00t among an online in-crowd, is a probably ephemeral variant.)
In the same vein, if you write it and it looks right in an echomimetic way, it probably will serve its purpose just fine, and only the most pedantic reader is going to lose focus and second guess you.
I like Ooo-wee Ooowee without the h. much like the way they spell the word Shawn as sene. Now sene sounds like sea Shawn to me is more proper. You're getting the sha sound for the name of Shawn.