library(dplyr)
library(tidyr)
> df
      Pedigree SNP1 SNP2
1 Individual 1    C    G
2 Individual 1    C    G
3 Individual 2    C    T
4 Individual 2    C    T

df %>% 
    mutate(type=rep(c("a1","a2"), count(.)/2)) %>% 
    pivot_longer(names_to = "SNP", -c(Pedigree,type))%>%
    pivot_wider(names_from = c(SNP,type),names_sep = ".", names_sort=T)

# A tibble: 2 x 5
  Pedigree     SNP1.a1 SNP1.a2 SNP2.a1 SNP2.a2
  <chr>        <chr>   <chr>   <chr>   <chr>  
1 Individual 1 C       C       G       G      
2 Individual 2 C       C       T       T

It's more like you need reshape2::dcast() here; however, you could also try:

df <- data.frame(
  SAM = c('Individual 1', 'Individual 1', 'Individual 2', 'Individual 2'),
  SNP1 = c('C', 'C', 'C', 'C'),
  SNP2 = c('G', 'G', 'T', 'T'))
df
           SAM SNP1 SNP2
1 Individual 1    C    G
2 Individual 1    C    G
3 Individual 2    C    T
4 Individual 2    C    T

samids <- unique(df$SAM)
new <- do.call(rbind, lapply(samids, function(x) unlist(df[df$SAM == x, -1])))
rownames(new) <- samids
new

             SNP11 SNP12 SNP21 SNP22
Individual 1 "C"   "C"   "G"   "G"  
Individual 2 "C"   "C"   "T"   "T"

Kevin