The list approach:
Clearly based on @Pascal 's idea. Not a fan of lists, but in some cases they are extremely helpful.
   set.seed(42)
x <- runif(100)
y <- 2*x+runif(100)
fit1 <- lm(y ~x)
set.seed(123)
x <- runif(100)
y <- 2*x+runif(100)
fit2 <- lm(y ~x)
# manually select model names
model_names = c("fit1","fit2")
# create a list based on models names provided
list_models = lapply(model_names, get)
# set names
names(list_models) = model_names
# check the output
list_models
# $fit1
# 
# Call:
#   lm(formula = y ~ x)
# 
# Coefficients:
#   (Intercept)            x  
#        0.5368       1.9678  
# 
# 
# $fit2
# 
# Call:
#   lm(formula = y ~ x)
# 
# Coefficients:
#   (Intercept)            x  
#        0.5545       1.9192 
Given that you have lots of models in your work space, the only "manual" thing you have to do is provide a vector of your models names (how are they stored) and then using the get function you can obtain the actual model objects with those names and save them in a list.
Store model objects in a dataset when you create them:
The data frame can be created using dplyr and do if you are planning to store the model objects when they are created. 
library(dplyr)
set.seed(42)
x1 = runif(100)
y1 = 2*x+runif(100)
set.seed(123)
x2 <- runif(100)
y2 <- 2*x+runif(100)
model_formulas = c("y1~x1", "y2~x2")
data.frame(model_formulas, stringsAsFactors = F) %>%
  group_by(model_formulas) %>%
  do(model = lm(.$model_formulas))
#     model_formulas   model
#              (chr)   (chr)
#   1          y1~x1 <S3:lm>
#   2          y2~x2 <S3:lm>
It REALLY depends on how "organised" is the process that allows you to built those 200+ models you mentioned. You can build your models this way if they depend on columns of a specific dataset. It will not work if you want to build models based on various columns of different datasets, maybe of different work spaces or different model types (linear/logistic regression). 
Store existing model objects in a dataset:
Actually I think you can still use dplyr using the same philosophy as in the list approach. If the models are already built you can use their names like this
library(dplyr)
set.seed(42)
x <- runif(100)
y <- 2*x+runif(100)
fit1 <- lm(y ~x)
set.seed(123)
x <- runif(100)
y <- 2*x+runif(100)
fit2 <- lm(y ~x)
# manually select model names
model_names = c("fit1","fit2")
data.frame(model_names, stringsAsFactors = F) %>%
  group_by(model_names) %>%
  do(model = get(.$model_names))
#   model_names   model
#         (chr)   (chr)
# 1        fit1 <S3:lm>
# 2        fit2 <S3:lm>