The Toffees were the only English top-flight club not to strengthen last month despite a desperate need to inject life into a squad that is second from bottom of the table with just three wins all season.
Numerous attempts were made to bring in players on Tuesday's deadline day, the day after Dyche was officially announced as manager, but Everton were beaten by rivals or could not get deals over the line.
Owner Farhad Moshiri had promised the club would sign a striker but Dyche said they had not failed for want of trying.
"Phone calls were made, the offers were made, it didn't get to them whatever the reason," the former Burnley boss said at his first press conference on Thursday.
"We can have debates on why (but) I'm not going to sign players for the sake of it. They've got to be good enough to affect what we do.
"There's good players here. They may not have shown it but we need to remind ourselves of the good players here. We've got to polish them up a bit."
Everton fans have been fiercely critical in recent weeks about the way the club is being run and the board was advised not to attend the recent home match against Southampton due to threats made over their safety.
Dyche, who faces a daunting first match against Premier League leaders Arsenal on Saturday, said he had walked into the situation with his eyes wide open but called for everyone to pull together to haul the club out of relegation trouble.
"The message is clear -- unity," he said. "It's a given. Even the most disgruntled Evertonians, give us a chance to stick by the team. They (fans) are important, players respond to fans and when the pressure lifts all of a sudden they feel the backing and support.
"I'm reaching out to them -- we'll give you honesty, work ethic. All we ask is give us a window to breathe, get ourselves going. Play your role in that."
Dyche was not Moshiri's first choice to replace the sacked Frank Lampard -- he wanted ex-Leeds boss Marcelo Bielsa but talks broke down -- and the 51-year-old admitted he split opinions among fans.
"Possibly if it (the club) wasn't in the shape it's in I wouldn't get the chance," he said. "I don't mind that. I'm a custodian. I won't be here forever but I'll look after it.
"I've got to earn my spurs and I'm willing to put the hard yards in to earn their respect."