Half the time, I’ll have my Bible at Mass, but it’s only because I have class before and we need our Bibles.

But, to answer the OP’s question, I would personally say this: “We don’t bring our own Bibles because we would be flipping pages too much to follow what we’re saying in the Mass!” And if they further inquire: “Well, just about everything we say in the Mass is Biblical, from the beginning to the end!” Which is true!
I’ve found passages like this in many Catholic books:
In the name of the Father, and of the Son, and of the Holy Spirit (Mt 28:19)
Amen (1 Chr 16:36)
The grace of our Lord Jesus Christ, the love of God, and the fellowship of the Holy Spirit be with you (2 Cor 13:14)
The Lord be with you (Ru 2:4)
Glory to God in the highest, and peace to His people on earth (Lk 2:14)
Blessed be God forever (Ps 68:36)
May the Lord accept the sacrifice at your hands (Ps 50:23)
Lift up your hearts; we lift them up to the Lord (Lam 3:41)
Let us give thanks to the Lord our God (Col 3:17)
Hosanna in the highest; blessed is He who comes in the name of the Lord (Mk 11:9-10)
Let us proclaim the mystery of our faith (1 Tm 3:16)
Christ has died, Christ is risen, Christ will come again (1 Cor 15:3-5; Rv 22:12)
Lord, you are holy indeed, the fountain of all holiness (2 Mc 14:36)
Through Him, with Him, in Him (Rom 11:36)
This is the Lamb of God; happy are those who are called to His supper (Rv 19:9)
And all the readings! There’s probably more, but that is what I found in Ask the Bible Geek by Mark Hart. I found a similar passage in the Lamb’s Supper, but a friend is borrowing that right now.

I always find it cool when I see lists like this, thought I’d share.