grep "^>" <filename> | sed 's/;/,/g' > <newfilename>

Command line answer. Grep will search for ">" and sed will substitute ";" with a tab creating new columns. the last ">" will output your results to the new file name you indicated.

Not a solution in R but you can simply do

$ grep "^>" your_file.fa | awk -F ">|;" '{for(i=2;i<NF;i++){printf "%s,", $i}; printf "\n"}'
Eukaryota,Alveolata,Dinoflagellata,Dinophyceae,Peridiniales,Kryptoperidiniaceae,Unruhdinium,Unruhdinium_kevei,

sed

 sed '/^>/s/;/\t/g;/^[^>]/d;s/^>//' in.fasta

Good bash solutions, that could be wrapped inside R for example as below:

library(data.table)
x <- fread("grep ... myFilename.fasta")

Or do all within R:

#example input fasta
x <- read.table(text = "
>seq0;x1;y1
FQTWEEFSRAAEKLYLADPMKVRVVLKYRHVDGNLCIKVTDDLVCLVYRTDQAQDVKKIEKF
>seq1;x22
KYRTWEEFTRAAEKLYQADPMKVRVVLKYRHCDGNLCIKVTDDVVCLLYRTDQAQDVKKIEKFHSQLMRLME
LKVTDNKECLKFKTDQAQEAKKMEKLNNIFFTLM
>seq2
EEYQTWEEFARAAEKLYLTDPMKVRVVLKYRHCDGNLCMKVTDDAVCLQYKTDQAQDVKKVEKLHGK
", sep = ";", fill = TRUE, header = FALSE)

# keep only header rows
x <- x[ grep("^>", x$V1),  ]

# remove ">"
x$V1 <- gsub(">", "", x$V1, fixed = TRUE)

# output
write.csv(x, "myFile.csv")

myFile.csv

"","V1","V2","V3"
"1","seq0","x1","y1"
"3","seq1","x22",""
"6","seq2","",""