Could it be part of "The Last Survivors" trilogy by Susan Beth Pfeffer?
The first book, Life as we knew it was published in 2006.
The description provided on Good Reads is:
Miranda’s disbelief turns to fear in a split second when a meteor
knocks the moon closer to the earth. How should her family prepare
for the future when worldwide tsunamis wipe out the coasts,
earthquakes rock the continents, and volcanic ash blocks out the sun?
As summer turns to Arctic winter, Miranda, her two brothers,
and their mother retreat to the unexpected safe haven of their
sunroom, where they subsist on stockpiled food and limited water in
the warmth of a wood-burning stove.
Told in journal entries, this is the heart-pounding story of Miranda’s
struggle to hold on to the most important resource of all--hope--in an
increasingly desperate and unfamiliar world.
- Moon is hit by a meteor.
- Moon comes closer to Earth
- Earth's climate changes as a result.
- Summer changes to arctic winter.
- Kids try to survive.
Told from POV of a girl named Myranda instead of a boy.
The girl has two brothers unlike your description where the boy has two sisters.
There is also the role of their mother.
The climate change is not limited to winter only
The second book in the series (The Dead and the Gone) may be a closer match as it follows the perspective of a boy and his 2 sisters in a city. From Wikipedia's synopsis:
The Dead and the Gone follows 17-year-old Alex Morales and his
sisters, Briana and Julie, in their struggle to survive after an
asteroid hits the Moon and knocks it out of orbit, closer to Earth.
Taking place in New York, they are plagued with volcanic eruptions,
tsunamis and tidal waves, and earthquakes, along with famine caused by
food shortages and disease that kill millions of people in the
process. Alex is forced to take care of his sisters in the absence of
his mother and father and to raid dead bodies for valuables to trade
for food. He struggles with his religious faith while trying
desperately to survive.