Using GNU sed we can clip any spaces sticking around a tabb as shown
$ sed -Ei -e 's/[ ]*\t[ ]*/\t/g' file
With awk we iterate over the fields and then trim the field:
$ awk -F '\t' -v OFS='\t' '
{
for (i=1; i<=NF; ++i) {
gsub(/^[ ]+|[ ]+$/, "", $i)
}
}1
' file > foo && mv foo file