Way to get genomic sequences at given coordinates without downloading fasta files of whole chromosomes/genomes first?
using a http request.

if there is a DAS server, you can always use this protocol to download the xml -> fasta. see https://www.biostars.org/p/56/

$ curl -s "http://genome.ucsc.edu/cgi-bin/das/hg19/dna?segment=chrM:100,200"  | xmllint --xpath '/DASDNA/SEQUENCE/DNA/text()' - | tr -d '\n'
ggagccggagcaccctatgtcgcagtatctgtctttgattcctgcctcattctattatttatcgcacctacgttcaatattacaggcgaacatacctacta


or use the UCSC utility twoBitToFa which works with remote files.

If you have an NCBI accession number you can use the E-utilities with seq_start and seq_end.

$ wget -q  -O -  "https://eutils.ncbi.nlm.nih.gov/entrez/eutils/efetch.fcgi?db=nuccore&amp;id=AE014134.1&amp;seq_start=100&amp;seq_stop=300&amp;rettype=fasta"
&gt;AE014134.1:100-300 Drosophila melanogaster chromosome 2L complete sequence
TGCCAACATATTGTGCTCTTTGATTTTTTGGCAACCCAAAATGGTGGCGGATGAACGAGATGATAATATA
TTCAAGTTGCCGCTAATCAGAAATAAATTCATTGCAACGTTAAATACAGCACAATATATGATCGCGTATG
CGAGAGTAGTGCCAACATATTGTGCTAATGAGTGCCTCTCGTTCTCTGTCTTATATTACCG