Can I use both Spearman's rank correlations and multiple linear regression on the same data? I have infant appetite trait variables that are ordinal and non-normally distributed, therefore I used Spearman's rank correlations to test their association with infant weight and BMI z-scores.
I want to go further and test the association between the appetite trait variables and infant weight and BMI z-score while controlling for other variables in the data set. Can I use multiple linear regression models to do so, or would it be inconsistent/wrong to use multiple linear regression with these data after I already decided nonparametric tests were required when doing simple correlations?

If you have ordinal outcome data, why not use ordinal regression? Frank Harrell has some useful links in this answer. The lrm() and orm() functions of his rms package provide tools (depending in part on how many outcome levels you have). This can be a good way to proceed even with continuous outcomes when you don't want to assume even spacing between outcome levels. The UCLA OARC website illustrates ordinal regression with a different tool available in R. The ordered-logit tag on this website will lead you to more information.
For ordinal predictors, it depends somewhat on how many levels there are. This page and its links have many suggestions.
If there are a few levels (up to 5 levels or so), then you can just treat each ordinal predictor as a multi-level categorical predictor in a regression model. With that few levels, if you think that each increase from level to level is equivalent, you could use orthogonal polynomial coding  for an ordinal predictor to evaluate linear, quadratic, and higher-order trends.
For more levels, it makes sense to keep an ordinal predictor as numeric but to fit outcomes smoothly against it, in a way that makes no assumption about each increase along the numeric ordinal scale having the same association with outcome. If there are so many levels that the ordinal values are effectively continuous, regression splines can be useful.
The R ordPens package has several tools for handling ordinal predictors in regression models. One is a way to do an ordinal ANOVA that might work well for evaluating a continuous outcome against an ordinal predictor. It also provides a way to include ordinal predictors in generalized additive models (GAM) fit via the R mgcv package. With a GAM you can perform regressions with nonlinear functions of predictors, functions that are learned from the data. See Section 7.7 of An Introduction to Statistical Learning, second edition.
