How do I extract values from a column based upon positive or negative values in another column?

is not accurate. Using a reprex. See the FAQ catches problems like this. There must have been something else going on in the session in which you produced the output shown.

library(dplyr)

d1 <- data.frame(
  ID = c("56789", "56789", "56789", "56789", "56789", "56789", "56789", "56789", "56789", "56789", "56789", "56789"), Book = c("Book_A", "Book_A", "Book_B", "Book_B", "Book_C", "Book_C", "Book_D", "Book_D", "Book_E", "Book_E", "Book_F", "Book_F"), Home = c("San Diego Padres", "San Diego Padres", "San Diego Padres", "San Diego Padres", "San Diego Padres", "San Diego Padres", "San Diego Padres", "San Diego Padres", "San Diego Padres", "San Diego Padres", "San Diego Padres", "San Diego Padres"),
  Away = c("Seattle Mariners", "Seattle Mariners", "Seattle Mariners", "Seattle Mariners", "Seattle Mariners", "Seattle Mariners", "Seattle Mariners", "Seattle Mariners", "Seattle Mariners", "Seattle Mariners", "Seattle Mariners", "Seattle Mariners"), Team = c("San Diego Padres", "Seattle Mariners", "San Diego Padres", "Seattle Mariners", "San Diego Padres", "Seattle Mariners", "San Diego Padres", "Seattle Mariners", "San Diego Padres", "Seattle Mariners", "San Diego Padres", "Seattle Mariners"), Price = c(133, -162, 125, -155, 130, -160, 130, -150, 130, -150, 130, -155), Points = c(-1.5, 1.5, -1.5, 1.5, -1.5, 1.5, -1.5, 1.5, -1.5, 1.5, -1.5, 1.5)
)

d2 <- data.frame(ID = c(
  "12345", "12345",
  "12345", "12345",
  "12345", "12345",
  "12345", "12345",
  "12345", "12345"
), Book = c(
  "Book_A", "Book_A", "Book_B",
  "Book_B", "Book_C", "Book_C", "Book_D", "Book_D", "Book_E",
  "Book_E"
), Home = c(
  "Cincinnati Reds", "Cincinnati Reds",
  "Cincinnati Reds", "Cincinnati Reds", "Cincinnati Reds", "Cincinnati Reds",
  "Cincinnati Reds", "Cincinnati Reds", "Cincinnati Reds", "Cincinnati Reds"
), Away = c(
  "Cleveland Guardians", "Cleveland Guardians", "Cleveland Guardians",
  "Cleveland Guardians", "Cleveland Guardians", "Cleveland Guardians", "Cleveland Guardians", "Cleveland Guardians", "Cleveland Guardians", "Cleveland Guardians"
), Team = c("Cincinnati Reds", "Cleveland Guardians", "Cincinnati Reds", "Cleveland Guardians", "Cincinnati Reds", "Cleveland Guardians", "Cincinnati Reds", "Cleveland Guardians", "Cincinnati Reds", "Cleveland Guardians"), Price = c(-175, 143, 160, -190, 140, -165, 145, -170, 150, -178), Points = c(1.5, -1.5, -1.5, 1.5, -1.5, 1.5, -1.5, 1.5, -1.5, 1.5))


d1 %>%
  group_by(ID, Team) %>% 
  slice_max(Price, with_ties = FALSE) %>% 
  group_by(ID) %>% 
  mutate(Value = function(Price[1], Price[2]))
# yields
#Error: unexpected '[' in:
#"  group_by(ID) %>% 
# mutate(Value = function(Price["
d2 %>%
  group_by(ID, Team) %>% 
  slice_max(Price, with_ties = FALSE) %>% 
  group_by(ID) %>% 
  mutate(Value = function(Price[1], Price[2]))
# same error

Combined with the lack of a clear description of of Value is derived, no path to a solution is apparent.