These little prepositions can be subtle. If you don't get them exactly right, people will usually know what you mean anyway, it just makes the difference between sounding completely natural and sounding a little unusual.
First of all, "She picked the ribbon." isn't right. "Picked", by itself, either means "selected; chose", or it's what you do to fruits & vegetables (picking peaches, etc.)
What she did to the ribbon was "pick it up". "She picked up the ribbon." sounds very idiomatic to me. Unless there is more to the story involving the ribbon, that one would be my choice.
I don't think I would say, "She took up the ribbon." "Took up" has a weighty connotation. You would use that expression for something that was either heavy or very important/significant:
"They took up their burdens and trudged along the dusty path."
"We must once again take up arms in the cause of freedom."
"He took up his hammer and began demolishing the structure."
I think ribbon is too lightweight for that expression.
"She took the ribbon." would be OK if she took the ribbon and kept it. If she stole it, she "took" it. Or if someone gave it to her, and she accepted it, then she "took" it.
On the other hand, even if she didn't keep it, as long as you tell what she did with it next, you can say "she took it":
"She took the ribbon and put it in her hair."
"She took the ribbon, then handed it to me."
"She took the ribbon and sniffed it."
I'm sure there are other prepositional subtleties that I haven't thought of here, but that's what comes to mind after thinking it over for a few minutes.
Bottom line: "She picked up the ribbon."