The Gunners have not won a trophy of any kind since 2005 but they are just a point off the pace in the early running this season, having reeled off three wins in a row.
And they secured a club-record 10th consecutive away victory when they triumphed 2-1 in the Champions League at Marseille on Wednesday.
Walcott said: "I think we'll push on and we can definitely challenge this year, for sure.
"There is so much belief in this team at the moment and I think it's showing. We just need to make sure we get that home form going now, the away form is absolutely fantastic.
"It's 10 successive wins away from home, a record. To be part of a team that has done that, in the amount of years that Arsenal have existed, is a great honour and a great achievement for everyone.
"It just shows how good and how talented these players are.
"We haven't got the biggest squad in the world and we've got a lot of injuries, which doesn't help. We've got to stay fit, we don't want to lose any more players.
"As soon as the players come back there is going to be more competition for places as well, that's only going to be a good thing for Arsenal."
Arsenal's Champions League group is tough, with fixtures against Napoli and Borussia Dortmund to come in October.
Regarding their victory in France, Walcott added: "It's a great start to the competition and we've got players who are playing fantastically well.
"We always felt we didn't want to lose, so to get the three points was an absolute bonus.
"We don't want to look too far ahead, we've got to concentrate on Sunday against Stoke first of all."