I discovered that the CDC keeps a record of every birth certificate in America going back to 1968 (at least the metadata). You can actually download all the metadata yourself, but the version they let you download redacts all the geographic (county) data. Which makes it kind of useless. BUT you can query the data for certain statistics through wonder.cdc.gov.

There are a few things I've been "wondering" about Mormons and their famously-high fertility since I saw this report which said the white fertility in Utah fell below replacement-level in 2017. So I queried the data for 3 representative geographies: the first obvious one being Utah (the Mormon Mothership) but Utah is pretty multicultural these days so you can't get an accurate picture of what's happening with the Mormons just by looking at Utah. Two much better geographies are Utah County, Utah (home of Brigham Young University) and Bonneville County, Idaho (neighbor of Brigham Young University - Idaho). These two yuge 99% Mormon universities make both of these counties nearly Mormon-monocultural so they should be great indicators of the current fertility rate in the Mormon community. Here are the results:

Edit 10:30PM MST: Thank you @PoisonAero for correcting the calculations! Now they are actually the Total Fertility Rate, and not the Crude Fertility Rate.
Edit 29 Oct 2021, 8:00PM MST: Updated charts adding 2020 Natality data

The fertilty rate of Utah County fell 24.7% from 3.04 in 2005 to 2.29 in 2020. The White (Non-Hispanic white) fertility rate of Utah County fell 24.8% from 2.98 in 2005 to 2.24 in 2020. The latter is now just barely above-replacement.

Sources:

Every value in the "Fertility Rate per 1,000" column of the following tables is an average annual fertility of women over a 5-year age range. To convert to Total Fertility Rate (TFR), sum the cohort-fertility rates together, multiply by 5, and divide by 1000.

Everyone:

(State of Utah)
2003-2006 https://wonder.cdc.gov/controller/saved/D27/D133F771
2007-2020 https://wonder.cdc.gov/controller/saved/D66/D133F772
(Counties)
2003-2006 https://wonder.cdc.gov/controller/saved/D27/D137F663
2007-2020 https://wonder.cdc.gov/controller/saved/D66/D137F669

Non-Hispanic Whites:

(State of Utah)
2003-2006 https://wonder.cdc.gov/controller/saved/D27/D243F692
2007-2015 https://wonder.cdc.gov/controller/saved/D66/D243F704
2016-2020 https://wonder.cdc.gov/controller/saved/D149/D243F699
(Counties)
2003-2006 https://wonder.cdc.gov/controller/saved/D27/D243F700
2007-2015 https://wonder.cdc.gov/controller/saved/D66/D243F708
2016-2020 https://wonder.cdc.gov/controller/saved/D149/D243F705

Addendum

Due to expressed interest, I have created a chart that shows the fertility rate of every county in Utah with available data (there were only 6 available counties). I have updated the source links as well; they now link to tables that include all available counties in Utah and Idaho.

If you compare to the SLTRIB chart that shows percent LDS for each county in Utah, you will see there is an almost 1-to-1 ordering between "percent LDS" and TFR.

Cohort fertility in Utah County

Utah County serves as a useful case study to see what is happening to fertility by age group. There have been large declines in cohort-fertility rate for 20-29 year olds while the cohort-fertility rate of 30-39 year olds has remained stable.