“I suppose so,” said Mrs. Dursley stiffly - (Harry Potter) [closed]

The answer to "Does it mean that Mrs. Dursley stands up?" is No and is correctly explained in the answer by user 66974. Therefore I won't repeat it. Instead I shall explain why it is not correct.

Adjective - stiff

Adverb - stiffly

The addition of "y" (as opposed to "ly") to an adjective, is a slang way to convert an adjective into a noun. Note that it is the added sound that is important, not the spelling.

stiffy, stiffie - A thing that is stiff, usually slang for erect penis
quicky, quickie - Something that is quick, usually slang for a short sex session
biggy, biggie - A thing that is big. E.g. There's a storm coming and I think it is going to be a biggie.
oldy, oldie - A thing that is old. E.g. When it comes to pop music, the oldies are the best.
fatty, fattie - A person who is fat (This is an insult)
meany, meanie - A person who is mean

When can this be used?

I think it would be very difficult to discover a rule. It might be possible but that would require a separate question. You cannot simply add "y" to any adjective of your choice. To form them requires a feel for the language.

For example, none of the following would work:

annoying ---> annoyingie, to mean an annoying person (incorrect)
fast ---> fasty, to mean something that goes fast (incorrect)
stupid ---> stupidie, to mean a stupid person (incorrect)

The root word here is "stiff", with the literal meaning "rigid; hard to bend; inflexible". It's quite a common word, with several figurative meanings and derived words more-or-less connected to that basic definition. Both "stiffy" and "stiffly" come from "stiff", but they are very different words, not connected to each other in any other way.

Adding a "y" or "ie" to the end of words is used in some slang (very informal) words to form a noun (a thing) from an adjective (a description of that thing). So the word you found, "stiffy", is a slang noun literally meaning "a stiff thing", with the particular meaning of an erect penis. An erect penis is rigid, so is literally stiff; there is no particular relevance to any other property like standing up or pointing upwards.

The word in the book is the very different "stiffly". The "ly" ending can be used to turn almost any adjective (describing a thing) into an adverb (describing an action), so "stiffly" basically means "in a stiff way". In this case, it's slightly figurative: Mrs Dursley is speaking unusually formally, or her manner is "stiff" as opposite to "relaxed".

To find the correct root in future, remember that adjectives that already end in "y" usually keep an extra syllable as adverbs, ending "ily" - for example "stuffy" becomes "stuffily, not "stuffly". (I'm sure there are exceptions to this rule, because in English there are always exceptions if you look hard enough!)