If you self-mill Creeping Chill and immediately shuffle it back with Stream of Thoughts, you can't choose to exile Creeping Chill, and you won't get its effect.
That is because Creeping Chill is a triggered ability rather than a replacement effect on a static ability.
603.1. Triggered abilities have a trigger condition and an effect. They are written as “[When/Whenever/At] [trigger condition or event], [effect]. [Instructions (if any).]”
The ability does not fit any replacement effect wording pattern:
614.1a Effects that use the word “instead” are replacement effects. Most replacement effects use the word “instead” to indicate what events will be replaced with other events.
As you correctly note, by the time Creeping Chill's ability resolves, the object it's referring to (the object represented by the card in the graveyard) is no longer in the expected zone (graveyard), because the card was already shuffled back into the library.
As for technicalities: First, Creeping Chill's ability triggers, but does not yet go on the stack while Stream of Thought is still resolving. It goes on the stack once SoT has finished resolving and the next player would get priority.
603.2. Whenever a game event or game state matches a triggered ability’s trigger event, that ability automatically triggers. The ability doesn’t do anything at this point.
603.3. Once an ability has triggered, its controller puts it on the stack as an object that’s not a card the next time a player would receive priority. See rule 117, “Timing and Priority.” The ability becomes the topmost object on the stack. It has the text of the ability that created it, and no other characteristics. It remains on the stack until it’s countered, it resolves, a rule causes it to be removed from the stack, or an effect moves it elsewhere.
Second, Creeping Chill's ability would not "fizzle". A spell or ability fizzles if all its targets become illegal between triggering and resolving. Since Creeping Chill's ability has no targets, they can't become illegal. The ability will resolve normally, you just can't choose to exile the card.
608.2b If the spell or ability specifies targets, it checks whether the targets are still legal. A target that’s no longer in the zone it was in when it was targeted is illegal. Other changes to the game state may cause a target to no longer be legal; for example, its characteristics may have changed or an effect may have changed the text of the spell. If the source of an ability has left the zone it was in, its last known information is used during this process. If all its targets, for every instance of the word “target,” are now illegal, the spell or ability doesn’t resolve. It’s removed from the stack and, if it’s a spell, put into its owner’s graveyard. Otherwise, the spell or ability will resolve normally. [..]
As an aside, if Creeping Chill's triggered ability was worded as a static ability with a replacement effect, it could be made to work with Stream of Thoughts.