Harry simply wasn't thinking straight or logically.
Harry was acting erratically and impulsively when he thought Sirius was in danger because of the visions the Dark Lord was giving him. He didn't see anything about that situation clearly because his judgment was clouded by his emotions. He also didn't think of using the two-way mirror Sirius gave him, even though Sirius at the time had specifically told him that it was something he could use if he needed him.
“What is it?’ Harry asked.
‘A way of letting me know if Snape’s giving you a hard time. No, don’t open it in here!’ said Sirius, with a wary look at Mrs Weasley, who was trying to persuade the twins to wear hand-knitted mittens. ‘I doubt Molly would approve – but I want you to use it if you need me, all right?’
‘OK,’ said Harry, stowing the package away in the inside pocket of his jacket, but he knew he would never use whatever it was. It would not be he, Harry, who lured Sirius from his place of safety, no matter how foully Snape treated him in their forthcoming Occlumency classes.”
- Harry Potter and the Order of the Phoenix, Chapter 24 (Occlumency)
Obviously, circumstances had changed since he decided that he wouldn't use the mirror - at the time he was sure that Sirius was already unsafe.
Hermione was handling shouting irrational Harry, she mightn't have time for lateral thinking.
Hermione was trying desperately to convince Harry not to just run off to the Ministry looking for Sirius, which is what he wanted to do. Once she got him to agree to try to contact Sirius first, she had to quickly think of a way to do it - Harry already wanted to just go to the Ministry and was already impatient and angry. It's likely that using the fireplace was the first form of communication she could think of, and at that point she didn't really have time to consider all of her options. She went right from convincing Harry to check on Sirius first to making a plan to do it.
“Sirius is being tortured NOW!’ shouted Harry. ‘We haven’t got time to waste.’
‘But if this is a trick of Voldemort’s, Harry, we’ve got to check, we’ve got to.’
‘How?’ Harry demanded. ‘How’re we going to check?’
‘We’ll have to use Umbridge’s fire and see if we can contact him,’ said Hermione, who looked positively terrified at the thought.”
- Harry Potter and the Order of the Phoenix, Chapter 32 (Out of the Fire)
In addition, Hermione knew Harry had used the fireplace to contact Sirius before, and so it was likely the first thing she thought of when trying to figure out how to contact him. It would likely be a lot easier for her to quickly think of something they'd used already than the indirect way of using the Room of Requirement, especially while trying to keep Harry from making reckless decisions.