What SMS apps exist on the iPhone that support ascii emoticons, not emoji?

Solution 1:

Apple does not allow SMS nor Phone replacement apps in the App Store, so this will never be possible without jailbreaking. Since you indicated you would prefer not to jailbreak, you will not be able to do this.

The closest you will be able to get would be to use a third-party SMS service which may either cost or perhaps force you into using a different phone number for SMS specifically with that app. You will need to review third-party apps available to see if they provide a custom keyboard to do what you wish to do.

Long story short: Built-in? Not possible outside of jmlumpkin's suggestion.