My dog is quite old, so I usually go on walks with him/her in a stroller.
My dog is quite old and can't really walk much, so I usually take him/her on walks in a stroller.
私の犬 - My dog
高齢 - very old, quite old
なので - so
バギー stroller, buggy (ペット用の物は pet stroller や dog stroller と言います。Pet buggyや dog buggy は私は聞いたことがありません)。子供用のもの(ベビーカー)は英語で buggy/ pram/ stroller/ pushchair と言います。
散歩する - go for a walk
私の犬は高齢なので、バギーに乗せて散歩しています - My dog is very old so I usually take walks with him/her in a stroller もしくは My dog is quite old so I usually go on walks with him/her in a stroller.
高齢犬があまり歩けないのでバギーに乗せて散歩しています - My dog is quite old and can't really walk much, so I usually go for walks with him/her in a stroller
自分のためじゃなくて犬のために散歩しているなら My dog is quite old and can't really walk much, so I usually take him/her on walks in a strollerのほうがいいです。