Dear John,
Allow me to use the biblical story of another couple in a similar situation (1 Samuel 1:1-18) to frame my advice.
In verse 8, Elkanah tries to console his childless wife, Hannah, by asking if he was not worth more to her than ten sons. Unfortunately, words like these that appeal to logic cannot change a person’s desires. Instead, sincerely acknowledge your wife’s pain without trying to change how she feels.
In verse 14, the priest Eli assumes that Hannah’s fevered prayer is the result of drunkenness. Similarly, we often assume we know the exact problem (and hence solution) the person is facing. While your wife is clearly distressed about being childless, her strong emotions suggest that more could be swirling beneath, such as what being childless means to her. When she is calm, invite her to share with you why this means so much to her.
In verse 10, we learn that Hannah intercedes directly with God, knowing that only he can grant her what she yearns for. If faith is part of your relationship, pray together about this, and truly surrender the outcome to God. To that end you may find it helpful to seek support from a pastor or trusted Christian friend. Hannah herself was no longer downcast after Eli exhorted her in verse 17 to go in peace, trusting that God has heard her prayer.
Finally, you mentioned the stress that this has caused your marriage. When your wife is not distressed, you may gently ask how you can tackle this together as a couple, including sharing in a non-judgemental way how this has been affecting you, as well as exploring other options such as adoption. Again, the support of a pastor, Christian friend or counsellor may be helpful if you find it hard to do this on your own.